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

Percentage Accurate: 99.7% → 99.7%

Time: 9.7s

Alternatives: 14

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/r* 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. metadata-eval 99.6%

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

   3. sqrt-prod 99.6%

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

   4. pow1/2 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

5. Applied egg-rr 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
6. Step-by-step derivation

   1. unpow1/2 99.6%

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

7. Simplified 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 73.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\sqrt{x \cdot 9}}\right)\right)} \]

   2. expm1-udef 73.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y}{\sqrt{x \cdot 9}}\right)} - 1\right)} \]

   3. div-inv 73.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \frac{1}{\sqrt{x \cdot 9}}}\right)} - 1\right) \]

   4. metadata-eval 73.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \left(e^{\mathsf{log1p}\left(y \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{x \cdot 9}}\right)} - 1\right) \]

   5. sqrt-div 73.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\sqrt{\frac{1}{x \cdot 9}}}\right)} - 1\right) \]

   6. metadata-eval 73.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \left(e^{\mathsf{log1p}\left(y \cdot \sqrt{\frac{1}{x \cdot \color{blue}{\frac{1}{0.1111111111111111}}}}\right)} - 1\right) \]

   7. div-inv 73.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \left(e^{\mathsf{log1p}\left(y \cdot \sqrt{\frac{1}{\color{blue}{\frac{x}{0.1111111111111111}}}}\right)} - 1\right) \]

   8. clear-num 73.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \left(e^{\mathsf{log1p}\left(y \cdot \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}}\right)} - 1\right) \]

9. Applied egg-rr 73.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \sqrt{\frac{0.1111111111111111}{x}}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 73.6%

       \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \sqrt{\frac{0.1111111111111111}{x}}\right)\right)} \]

    2. expm1-log1p 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+66} \lor \neg \left(y \leq 1.65 \cdot 10^{+65}\right):\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.3e+66) (not (<= y 1.65e+65)))
   (+ 1.0 (* y (* -0.3333333333333333 (pow x -0.5))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.3e+66) || !(y <= 1.65e+65)) {
		tmp = 1.0 + (y * (-0.3333333333333333 * pow(x, -0.5)));
	} 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 <= (-5.3d+66)) .or. (.not. (y <= 1.65d+65))) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) * (x ** (-0.5d0))))
    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 <= -5.3e+66) || !(y <= 1.65e+65)) {
		tmp = 1.0 + (y * (-0.3333333333333333 * Math.pow(x, -0.5)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.3e+66) or not (y <= 1.65e+65):
		tmp = 1.0 + (y * (-0.3333333333333333 * math.pow(x, -0.5)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.3e+66) || !(y <= 1.65e+65))
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5))));
	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 <= -5.3e+66) || ~((y <= 1.65e+65)))
		tmp = 1.0 + (y * (-0.3333333333333333 * (x ^ -0.5)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.3e+66], N[Not[LessEqual[y, 1.65e+65]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $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 < -5.2999999999999997e66 or 1.65000000000000012e65 < y

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. neg-mul-1 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r/ 99.3%

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

      9. fma-def 99.3%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.8%

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

      1. associate-*r* 94.9%

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

   6. Simplified 94.9%

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

      1. pow1/2 94.9%

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

      2. inv-pow 94.9%

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

      3. pow-pow 94.8%

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

      4. metadata-eval 94.8%

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

      5. expm1-log1p-u 92.7%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

      6. expm1-udef 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. expm1-def 92.7%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

      2. expm1-log1p 94.8%

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

   10. Simplified 94.8%

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

       1. pow1 94.8%

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

       2. associate-*l* 94.8%

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

   12. Applied egg-rr 94.8%

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

       1. unpow1 94.8%

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

       2. *-commutative 94.8%

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

       3. associate-*l* 94.8%

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

   14. Simplified 94.8%

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

   if -5.2999999999999997e66 < y < 1.65000000000000012e65

   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. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.8%

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

      14. distribute-neg-frac 99.8%

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

      15. metadata-eval 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.0%

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

      3. frac-times 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

      6. frac-times 54.0%

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

      7. pow1 54.0%

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

      8. pow1 54.0%

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

      9. pow-prod-up 54.0%

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

      10. metadata-eval 54.0%

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

      11. associate-/r* 54.0%

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

      12. *-commutative 54.0%

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

      13. metadata-eval 54.0%

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

      14. pow2 54.0%

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

      15. sqrt-unprod 54.1%

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

      16. add-sqr-sqrt 54.1%

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

      17. inv-pow 54.1%

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

   6. Applied egg-rr 54.1%

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

      1. unpow-1 54.1%

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

      2. *-commutative 54.1%

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

      3. associate-/r* 54.1%

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

      4. metadata-eval 54.1%

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

      5. metadata-eval 54.1%

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

      6. associate-*r/ 54.1%

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

      7. clear-num 54.1%

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

      8. un-div-inv 54.1%

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

      9. clear-num 54.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 72.9%

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

      12. sqr-neg 72.9%

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

      13. mul-1-neg 72.9%

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

      14. mul-1-neg 72.9%

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

      15. sqrt-unprod 97.3%

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

      16. add-sqr-sqrt 97.4%

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

      17. associate-*r/ 97.4%

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

      18. metadata-eval 97.4%

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

      19. clear-num 97.4%

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

      20. div-inv 97.5%

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

      21. metadata-eval 97.5%

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

   8. Applied egg-rr 97.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+66} \lor \neg \left(y \leq 1.65 \cdot 10^{+65}\right):\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 3: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+66}:\\ \;\;\;\;1 + \left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+66}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e+66)
   (+ 1.0 (* (* y -0.3333333333333333) (sqrt (/ 1.0 x))))
   (if (<= y 1.7e+66)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (+ 1.0 (* y (* -0.3333333333333333 (pow x -0.5)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+66) {
		tmp = 1.0 + ((y * -0.3333333333333333) * sqrt((1.0 / x)));
	} else if (y <= 1.7e+66) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 * pow(x, -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.65d+66)) then
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) * sqrt((1.0d0 / x)))
    else if (y <= 1.7d+66) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) * (x ** (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+66) {
		tmp = 1.0 + ((y * -0.3333333333333333) * Math.sqrt((1.0 / x)));
	} else if (y <= 1.7e+66) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 * Math.pow(x, -0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.65e+66:
		tmp = 1.0 + ((y * -0.3333333333333333) * math.sqrt((1.0 / x)))
	elif y <= 1.7e+66:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 * math.pow(x, -0.5)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65e+66)
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) * sqrt(Float64(1.0 / x))));
	elseif (y <= 1.7e+66)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.65e+66)
		tmp = 1.0 + ((y * -0.3333333333333333) * sqrt((1.0 / x)));
	elseif (y <= 1.7e+66)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 + (y * (-0.3333333333333333 * (x ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65e+66], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+66], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6500000000000001e66

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. neg-mul-1 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r/ 99.2%

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

      9. fma-def 99.3%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 95.3%

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

      1. associate-*r* 95.5%

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

   6. Simplified 95.5%

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

   if -1.6500000000000001e66 < y < 1.70000000000000015e66

   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. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.8%

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

      14. distribute-neg-frac 99.8%

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

      15. metadata-eval 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.0%

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

      3. frac-times 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

      6. frac-times 54.0%

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

      7. pow1 54.0%

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

      8. pow1 54.0%

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

      9. pow-prod-up 54.0%

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

      10. metadata-eval 54.0%

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

      11. associate-/r* 54.0%

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

      12. *-commutative 54.0%

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

      13. metadata-eval 54.0%

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

      14. pow2 54.0%

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

      15. sqrt-unprod 54.1%

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

      16. add-sqr-sqrt 54.1%

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

      17. inv-pow 54.1%

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

   6. Applied egg-rr 54.1%

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

      1. unpow-1 54.1%

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

      2. *-commutative 54.1%

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

      3. associate-/r* 54.1%

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

      4. metadata-eval 54.1%

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

      5. metadata-eval 54.1%

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

      6. associate-*r/ 54.1%

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

      7. clear-num 54.1%

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

      8. un-div-inv 54.1%

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

      9. clear-num 54.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 72.9%

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

      12. sqr-neg 72.9%

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

      13. mul-1-neg 72.9%

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

      14. mul-1-neg 72.9%

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

      15. sqrt-unprod 97.3%

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

      16. add-sqr-sqrt 97.4%

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

      17. associate-*r/ 97.4%

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

      18. metadata-eval 97.4%

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

      19. clear-num 97.4%

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

      20. div-inv 97.5%

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

      21. metadata-eval 97.5%

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

   8. Applied egg-rr 97.5%

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

   if 1.70000000000000015e66 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-neg-frac 99.5%

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

      6. neg-mul-1 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*r/ 99.4%

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

      9. fma-def 99.4%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.5%

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

      14. distribute-neg-frac 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 94.2%

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

      1. associate-*r* 94.2%

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

   6. Simplified 94.2%

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

      1. pow1/2 94.2%

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

      2. inv-pow 94.2%

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

      3. pow-pow 94.2%

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

      4. metadata-eval 94.2%

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

      5. expm1-log1p-u 91.9%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

      6. expm1-udef 63.9%

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

   8. Applied egg-rr 63.9%

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

      1. expm1-def 91.9%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

      2. expm1-log1p 94.2%

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

   10. Simplified 94.2%

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

       1. pow1 94.2%

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

       2. associate-*l* 94.1%

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

   12. Applied egg-rr 94.1%

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

       1. unpow1 94.1%

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

       2. *-commutative 94.1%

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

       3. associate-*l* 94.3%

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

   14. Simplified 94.3%

       \[\leadsto 1 + \color{blue}{y \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+66}:\\ \;\;\;\;1 + \left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+66}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+67} \lor \neg \left(y \leq 3.1 \cdot 10^{+64}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.4e+67) (not (<= y 3.1e+64)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e+67) || !(y <= 3.1e+64)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.4d+67)) .or. (.not. (y <= 3.1d+64))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e+67) || !(y <= 3.1e+64)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.4e+67) or not (y <= 3.1e+64):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.4e+67) || !(y <= 3.1e+64))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.4e+67) || ~((y <= 3.1e+64)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.4e+67], N[Not[LessEqual[y, 3.1e+64]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999e67 or 3.0999999999999999e64 < y

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. neg-mul-1 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r/ 99.3%

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

      9. fma-def 99.3%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.8%

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

      1. associate-*r* 94.9%

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

   6. Simplified 94.9%

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

      1. expm1-log1p-u 46.6%

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

      2. expm1-udef 46.6%

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

      3. log1p-udef 46.6%

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

      4. add-exp-log 94.9%

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

      5. +-commutative 94.9%

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

      6. associate-*l* 94.8%

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

      7. fma-def 94.8%

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

      8. sqrt-div 94.7%

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

      9. metadata-eval 94.7%

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

      10. un-div-inv 94.7%

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

   8. Applied egg-rr 94.7%

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

      1. fma-udef 94.7%

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

      2. associate--l+ 94.7%

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

      3. metadata-eval 94.7%

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

      4. +-rgt-identity 94.7%

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

   10. Simplified 94.7%

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

   if -1.3999999999999999e67 < y < 3.0999999999999999e64

   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. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.8%

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

      14. distribute-neg-frac 99.8%

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

      15. metadata-eval 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.0%

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

      3. frac-times 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

      6. frac-times 54.0%

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

      7. pow1 54.0%

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

      8. pow1 54.0%

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

      9. pow-prod-up 54.0%

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

      10. metadata-eval 54.0%

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

      11. associate-/r* 54.0%

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

      12. *-commutative 54.0%

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

      13. metadata-eval 54.0%

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

      14. pow2 54.0%

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

      15. sqrt-unprod 54.1%

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

      16. add-sqr-sqrt 54.1%

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

      17. inv-pow 54.1%

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

   6. Applied egg-rr 54.1%

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

      1. unpow-1 54.1%

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

      2. *-commutative 54.1%

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

      3. associate-/r* 54.1%

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

      4. metadata-eval 54.1%

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

      5. metadata-eval 54.1%

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

      6. associate-*r/ 54.1%

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

      7. clear-num 54.1%

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

      8. un-div-inv 54.1%

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

      9. clear-num 54.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 72.9%

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

      12. sqr-neg 72.9%

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

      13. mul-1-neg 72.9%

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

      14. mul-1-neg 72.9%

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

      15. sqrt-unprod 97.3%

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

      16. add-sqr-sqrt 97.4%

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

      17. associate-*r/ 97.4%

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

      18. metadata-eval 97.4%

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

      19. clear-num 97.4%

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

      20. div-inv 97.5%

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

      21. metadata-eval 97.5%

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

   8. Applied egg-rr 97.5%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+67} \lor \neg \left(y \leq 3.1 \cdot 10^{+64}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 5: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+66} \lor \neg \left(y \leq 5.8 \cdot 10^{+64}\right):\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.8e+66) (not (<= y 5.8e+64)))
   (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e+66) || !(y <= 5.8e+64)) {
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.8d+66)) .or. (.not. (y <= 5.8d+64))) then
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.8e+66) || !(y <= 5.8e+64)) {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.8e+66) or not (y <= 5.8e+64):
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.8e+66) || !(y <= 5.8e+64))
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.8e+66) || ~((y <= 5.8e+64)))
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.8e+66], N[Not[LessEqual[y, 5.8e+64]], $MachinePrecision]], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8000000000000007e66 or 5.79999999999999986e64 < y

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. neg-mul-1 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r/ 99.3%

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

      9. fma-def 99.3%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 94.8%

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

      1. associate-*r* 94.9%

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

   6. Simplified 94.9%

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

      1. sqrt-div 94.6%

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

      2. metadata-eval 94.6%

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

      3. un-div-inv 94.8%

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

   8. Applied egg-rr 94.8%

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

   if -7.8000000000000007e66 < y < 5.79999999999999986e64

   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. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. fma-def 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-/r* 99.8%

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

      14. distribute-neg-frac 99.8%

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

      15. metadata-eval 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.0%

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

      3. frac-times 54.0%

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

      4. metadata-eval 54.0%

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

      5. metadata-eval 54.0%

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

      6. frac-times 54.0%

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

      7. pow1 54.0%

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

      8. pow1 54.0%

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

      9. pow-prod-up 54.0%

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

      10. metadata-eval 54.0%

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

      11. associate-/r* 54.0%

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

      12. *-commutative 54.0%

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

      13. metadata-eval 54.0%

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

      14. pow2 54.0%

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

      15. sqrt-unprod 54.1%

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

      16. add-sqr-sqrt 54.1%

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

      17. inv-pow 54.1%

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

   6. Applied egg-rr 54.1%

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

      1. unpow-1 54.1%

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

      2. *-commutative 54.1%

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

      3. associate-/r* 54.1%

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

      4. metadata-eval 54.1%

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

      5. metadata-eval 54.1%

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

      6. associate-*r/ 54.1%

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

      7. clear-num 54.1%

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

      8. un-div-inv 54.1%

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

      9. clear-num 54.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 72.9%

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

      12. sqr-neg 72.9%

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

      13. mul-1-neg 72.9%

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

      14. mul-1-neg 72.9%

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

      15. sqrt-unprod 97.3%

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

      16. add-sqr-sqrt 97.4%

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

      17. associate-*r/ 97.4%

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

      18. metadata-eval 97.4%

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

      19. clear-num 97.4%

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

      20. div-inv 97.5%

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

      21. metadata-eval 97.5%

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

   8. Applied egg-rr 97.5%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+66} \lor \neg \left(y \leq 5.8 \cdot 10^{+64}\right):\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

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(1.0 + Float64(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[(1.0 + N[(N[(-0.1111111111111111 / x), $MachinePrecision] + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate--l- 99.6%

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

   2. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-neg-frac 99.6%

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

   6. neg-mul-1 99.6%

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

   7. *-commutative 99.6%

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

   8. associate-*r/ 99.6%

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

   9. fma-def 99.6%

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

   10. associate-/r* 99.7%

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

   11. metadata-eval 99.7%

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

   12. *-commutative 99.7%

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

   13. associate-/r* 99.6%

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

   14. distribute-neg-frac 99.6%

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

   15. metadata-eval 99.6%

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

   16. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*r/ 99.6%

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

   4. associate-*l/ 99.6%

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

   5. *-commutative 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 7: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. 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. +-commutative 99.6%

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

   4. associate-/r* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 99.6%

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

5. Final simplification 99.6%

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

Alternative 8: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.012345679012345678}{x \cdot x}\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{+108}:\\ \;\;\;\;1 + \sqrt{t_0}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + t_0}{\frac{-0.1111111111111111}{x} + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 0.012345679012345678 (* x x))))
   (if (<= y -6.6e+108)
     (+ 1.0 (sqrt t_0))
     (if (<= y 1.35e+154)
       (+ 1.0 (/ -1.0 (* x 9.0)))
       (/ (+ -1.0 t_0) (+ (/ -0.1111111111111111 x) -1.0))))))

C

double code(double x, double y) {
	double t_0 = 0.012345679012345678 / (x * x);
	double tmp;
	if (y <= -6.6e+108) {
		tmp = 1.0 + sqrt(t_0);
	} else if (y <= 1.35e+154) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (-1.0 + t_0) / ((-0.1111111111111111 / x) + -1.0);
	}
	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 = 0.012345679012345678d0 / (x * x)
    if (y <= (-6.6d+108)) then
        tmp = 1.0d0 + sqrt(t_0)
    else if (y <= 1.35d+154) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = ((-1.0d0) + t_0) / (((-0.1111111111111111d0) / x) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.012345679012345678 / (x * x);
	double tmp;
	if (y <= -6.6e+108) {
		tmp = 1.0 + Math.sqrt(t_0);
	} else if (y <= 1.35e+154) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (-1.0 + t_0) / ((-0.1111111111111111 / x) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.012345679012345678 / (x * x)
	tmp = 0
	if y <= -6.6e+108:
		tmp = 1.0 + math.sqrt(t_0)
	elif y <= 1.35e+154:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (-1.0 + t_0) / ((-0.1111111111111111 / x) + -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.012345679012345678 / Float64(x * x))
	tmp = 0.0
	if (y <= -6.6e+108)
		tmp = Float64(1.0 + sqrt(t_0));
	elseif (y <= 1.35e+154)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(-1.0 + t_0) / Float64(Float64(-0.1111111111111111 / x) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.012345679012345678 / (x * x);
	tmp = 0.0;
	if (y <= -6.6e+108)
		tmp = 1.0 + sqrt(t_0);
	elseif (y <= 1.35e+154)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (-1.0 + t_0) / ((-0.1111111111111111 / x) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.6e+108], N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + t$95$0), $MachinePrecision] / N[(N[(-0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.60000000000000038e108

   1. Initial program 99.3%

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

      1. associate--l- 99.3%

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

      2. sub-neg 99.3%

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

      3. +-commutative 99.3%

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

      4. distribute-neg-in 99.3%

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

      5. distribute-neg-frac 99.3%

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

      6. neg-mul-1 99.3%

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

      7. *-commutative 99.3%

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

      8. associate-*r/ 99.1%

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

      9. fma-def 99.2%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 2.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 16.8%

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

      3. frac-times 16.8%

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

      4. metadata-eval 16.8%

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

   6. Applied egg-rr 16.8%

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

   if -6.60000000000000038e108 < y < 1.35000000000000003e154

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 88.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 48.0%

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

      3. frac-times 48.0%

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

      4. metadata-eval 48.0%

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

      5. metadata-eval 48.0%

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

      6. frac-times 48.0%

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

      7. pow1 48.0%

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

      8. pow1 48.0%

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

      9. pow-prod-up 48.0%

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

      10. metadata-eval 48.0%

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

      11. associate-/r* 48.0%

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

      12. *-commutative 48.0%

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

      13. metadata-eval 48.0%

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

      14. pow2 48.0%

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

      15. sqrt-unprod 48.1%

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

      16. add-sqr-sqrt 48.1%

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

      17. inv-pow 48.1%

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

   6. Applied egg-rr 48.1%

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

      1. unpow-1 48.1%

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

      2. *-commutative 48.1%

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

      3. associate-/r* 48.1%

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

      4. metadata-eval 48.1%

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

      5. metadata-eval 48.1%

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

      6. associate-*r/ 48.1%

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

      7. clear-num 48.1%

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

      8. un-div-inv 48.1%

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

      9. clear-num 48.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 64.8%

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

      12. sqr-neg 64.8%

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

      13. mul-1-neg 64.8%

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

      14. mul-1-neg 64.8%

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

      15. sqrt-unprod 87.9%

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

      16. add-sqr-sqrt 88.0%

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

      17. associate-*r/ 88.0%

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

      18. metadata-eval 88.0%

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

      19. clear-num 88.0%

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

      20. div-inv 88.1%

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

      21. metadata-eval 88.1%

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

   8. Applied egg-rr 88.1%

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

   if 1.35000000000000003e154 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*r/ 99.5%

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

      9. fma-def 99.5%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 4.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.5%

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

      3. frac-times 0.5%

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

      4. metadata-eval 0.5%

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

      5. metadata-eval 0.5%

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

      6. frac-times 0.5%

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

      7. pow1 0.5%

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

      8. pow1 0.5%

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

      9. pow-prod-up 0.5%

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

      10. metadata-eval 0.5%

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

      11. associate-/r* 0.5%

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

      12. *-commutative 0.5%

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

      13. metadata-eval 0.5%

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

      14. pow2 0.5%

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

      15. sqrt-unprod 0.6%

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

      16. add-sqr-sqrt 0.6%

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

      17. frac-2neg 0.6%

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

      18. metadata-eval 0.6%

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

      19. div-inv 0.6%

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

      20. distribute-rgt-neg-in 0.6%

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

      21. metadata-eval 0.6%

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

      22. metadata-eval 0.6%

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

      23. div-inv 0.6%

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

      24. clear-num 0.6%

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

   6. Applied egg-rr 0.6%

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

      1. +-commutative 0.6%

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

      2. flip-+ 0.5%

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

      3. mul-1-neg 0.5%

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

      4. mul-1-neg 0.5%

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

      5. sqr-neg 0.5%

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

      6. div-inv 0.5%

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

      7. div-inv 0.5%

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

      8. swap-sqr 0.5%

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

      9. metadata-eval 0.5%

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

      10. inv-pow 0.5%

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

      11. inv-pow 0.5%

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

      12. pow-prod-up 0.5%

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

      13. metadata-eval 0.5%

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

      14. metadata-eval 0.5%

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

   8. Applied egg-rr 29.0%

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

   9. Taylor expanded in x around 0 29.0%

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

       1. unpow2 29.0%

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

   11. Simplified 29.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+108}:\\ \;\;\;\;1 + \sqrt{\frac{0.012345679012345678}{x \cdot x}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.012345679012345678}{x \cdot x}}{\frac{-0.1111111111111111}{x} + -1}\\ \end{array} \]

Alternative 9: 67.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.1111111111111111}{x} + -1\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{-1 + \frac{-0.1111111111111111}{x \cdot \left(x \cdot 9\right)}}{t_0}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.012345679012345678}{x \cdot x}}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ -0.1111111111111111 x) -1.0)))
   (if (<= y -3.1e+108)
     (/ (+ -1.0 (/ -0.1111111111111111 (* x (* x 9.0)))) t_0)
     (if (<= y 1.35e+154)
       (+ 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 = (-0.1111111111111111 / x) + -1.0;
	double tmp;
	if (y <= -3.1e+108) {
		tmp = (-1.0 + (-0.1111111111111111 / (x * (x * 9.0)))) / t_0;
	} else if (y <= 1.35e+154) {
		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 = ((-0.1111111111111111d0) / x) + (-1.0d0)
    if (y <= (-3.1d+108)) then
        tmp = ((-1.0d0) + ((-0.1111111111111111d0) / (x * (x * 9.0d0)))) / t_0
    else if (y <= 1.35d+154) 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 = (-0.1111111111111111 / x) + -1.0;
	double tmp;
	if (y <= -3.1e+108) {
		tmp = (-1.0 + (-0.1111111111111111 / (x * (x * 9.0)))) / t_0;
	} else if (y <= 1.35e+154) {
		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 = (-0.1111111111111111 / x) + -1.0
	tmp = 0
	if y <= -3.1e+108:
		tmp = (-1.0 + (-0.1111111111111111 / (x * (x * 9.0)))) / t_0
	elif y <= 1.35e+154:
		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(Float64(-0.1111111111111111 / x) + -1.0)
	tmp = 0.0
	if (y <= -3.1e+108)
		tmp = Float64(Float64(-1.0 + Float64(-0.1111111111111111 / Float64(x * Float64(x * 9.0)))) / t_0);
	elseif (y <= 1.35e+154)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(-1.0 + Float64(0.012345679012345678 / Float64(x * x))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (-0.1111111111111111 / x) + -1.0;
	tmp = 0.0;
	if (y <= -3.1e+108)
		tmp = (-1.0 + (-0.1111111111111111 / (x * (x * 9.0)))) / t_0;
	elseif (y <= 1.35e+154)
		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[(N[(-0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, -3.1e+108], N[(N[(-1.0 + N[(-0.1111111111111111 / N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 1.35e+154], 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 < -3.1000000000000001e108

   1. Initial program 99.3%

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

      1. associate--l- 99.3%

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

      2. sub-neg 99.3%

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

      3. +-commutative 99.3%

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

      4. distribute-neg-in 99.3%

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

      5. distribute-neg-frac 99.3%

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

      6. neg-mul-1 99.3%

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

      7. *-commutative 99.3%

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

      8. associate-*r/ 99.1%

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

      9. fma-def 99.2%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. distribute-neg-frac 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 2.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 16.8%

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

      3. frac-times 16.8%

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

      4. metadata-eval 16.8%

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

      5. metadata-eval 16.8%

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

      6. frac-times 16.8%

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

      7. pow1 16.8%

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

      8. pow1 16.8%

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

      9. pow-prod-up 16.8%

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

      10. metadata-eval 16.8%

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

      11. associate-/r* 16.8%

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

      12. *-commutative 16.8%

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

      13. metadata-eval 16.8%

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

      14. pow2 16.8%

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

      15. sqrt-unprod 5.9%

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

      16. add-sqr-sqrt 5.9%

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

      17. frac-2neg 5.9%

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

      18. metadata-eval 5.9%

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

      19. div-inv 5.9%

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

      20. distribute-rgt-neg-in 5.9%

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

      21. metadata-eval 5.9%

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

      22. metadata-eval 5.9%

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

      23. div-inv 5.9%

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

      24. clear-num 5.9%

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

   6. Applied egg-rr 5.9%

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

      1. +-commutative 5.9%

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

      2. flip-+ 16.8%

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

      3. mul-1-neg 16.8%

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

      4. mul-1-neg 16.8%

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

      5. sqr-neg 16.8%

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

      6. div-inv 16.8%

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

      7. div-inv 16.8%

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

      8. swap-sqr 16.8%

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

      9. metadata-eval 16.8%

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

      10. inv-pow 16.8%

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

      11. inv-pow 16.8%

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

      12. pow-prod-up 16.8%

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

      13. metadata-eval 16.8%

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

      14. metadata-eval 16.8%

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

   8. Applied egg-rr 2.9%

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

      1. metadata-eval 2.9%

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

      2. sqr-pow 2.9%

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

      3. metadata-eval 2.9%

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

      4. inv-pow 2.9%

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

      5. metadata-eval 2.9%

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

      6. inv-pow 2.9%

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

      7. swap-sqr 2.9%

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

      8. div-inv 2.9%

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

      9. div-inv 2.9%

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

      10. clear-num 2.9%

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

      11. frac-times 2.9%

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

      12. metadata-eval 2.9%

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

      13. clear-num 2.9%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 16.8%

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

      16. frac-times 16.8%

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

      17. metadata-eval 16.8%

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

      18. metadata-eval 16.8%

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

      19. frac-times 16.8%

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

      20. sqrt-unprod 16.8%

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

      21. add-sqr-sqrt 16.8%

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

      22. clear-num 16.8%

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

      23. div-inv 16.8%

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

      24. metadata-eval 16.8%

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

   10. Applied egg-rr 16.8%

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

   if -3.1000000000000001e108 < y < 1.35000000000000003e154

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 88.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 48.0%

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

      3. frac-times 48.0%

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

      4. metadata-eval 48.0%

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

      5. metadata-eval 48.0%

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

      6. frac-times 48.0%

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

      7. pow1 48.0%

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

      8. pow1 48.0%

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

      9. pow-prod-up 48.0%

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

      10. metadata-eval 48.0%

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

      11. associate-/r* 48.0%

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

      12. *-commutative 48.0%

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

      13. metadata-eval 48.0%

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

      14. pow2 48.0%

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

      15. sqrt-unprod 48.1%

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

      16. add-sqr-sqrt 48.1%

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

      17. inv-pow 48.1%

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

   6. Applied egg-rr 48.1%

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

      1. unpow-1 48.1%

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

      2. *-commutative 48.1%

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

      3. associate-/r* 48.1%

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

      4. metadata-eval 48.1%

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

      5. metadata-eval 48.1%

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

      6. associate-*r/ 48.1%

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

      7. clear-num 48.1%

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

      8. un-div-inv 48.1%

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

      9. clear-num 48.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 64.8%

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

      12. sqr-neg 64.8%

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

      13. mul-1-neg 64.8%

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

      14. mul-1-neg 64.8%

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

      15. sqrt-unprod 87.9%

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

      16. add-sqr-sqrt 88.0%

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

      17. associate-*r/ 88.0%

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

      18. metadata-eval 88.0%

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

      19. clear-num 88.0%

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

      20. div-inv 88.1%

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

      21. metadata-eval 88.1%

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

   8. Applied egg-rr 88.1%

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

   if 1.35000000000000003e154 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*r/ 99.5%

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

      9. fma-def 99.5%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r* 99.7%

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

      14. distribute-neg-frac 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 4.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.5%

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

      3. frac-times 0.5%

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

      4. metadata-eval 0.5%

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

      5. metadata-eval 0.5%

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

      6. frac-times 0.5%

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

      7. pow1 0.5%

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

      8. pow1 0.5%

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

      9. pow-prod-up 0.5%

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

      10. metadata-eval 0.5%

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

      11. associate-/r* 0.5%

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

      12. *-commutative 0.5%

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

      13. metadata-eval 0.5%

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

      14. pow2 0.5%

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

      15. sqrt-unprod 0.6%

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

      16. add-sqr-sqrt 0.6%

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

      17. frac-2neg 0.6%

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

      18. metadata-eval 0.6%

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

      19. div-inv 0.6%

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

      20. distribute-rgt-neg-in 0.6%

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

      21. metadata-eval 0.6%

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

      22. metadata-eval 0.6%

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

      23. div-inv 0.6%

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

      24. clear-num 0.6%

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

   6. Applied egg-rr 0.6%

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

      1. +-commutative 0.6%

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

      2. flip-+ 0.5%

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

      3. mul-1-neg 0.5%

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

      4. mul-1-neg 0.5%

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

      5. sqr-neg 0.5%

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

      6. div-inv 0.5%

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

      7. div-inv 0.5%

         \[\leadsto \frac{\left(-0.1111111111111111 \cdot \frac{1}{x}\right) \cdot \color{blue}{\left(-0.1111111111111111 \cdot \frac{1}{x}\right)} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      8. swap-sqr 0.5%

         \[\leadsto \frac{\color{blue}{\left(-0.1111111111111111 \cdot -0.1111111111111111\right) \cdot \left(\frac{1}{x} \cdot \frac{1}{x}\right)} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      9. metadata-eval 0.5%

         \[\leadsto \frac{\color{blue}{0.012345679012345678} \cdot \left(\frac{1}{x} \cdot \frac{1}{x}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      10. inv-pow 0.5%

          \[\leadsto \frac{0.012345679012345678 \cdot \left(\color{blue}{{x}^{-1}} \cdot \frac{1}{x}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      11. inv-pow 0.5%

          \[\leadsto \frac{0.012345679012345678 \cdot \left({x}^{-1} \cdot \color{blue}{{x}^{-1}}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      12. pow-prod-up 0.5%

          \[\leadsto \frac{0.012345679012345678 \cdot \color{blue}{{x}^{\left(-1 + -1\right)}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      13. metadata-eval 0.5%

          \[\leadsto \frac{0.012345679012345678 \cdot {x}^{\color{blue}{-2}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      14. metadata-eval 0.5%

          \[\leadsto \frac{0.012345679012345678 \cdot {x}^{-2} - \color{blue}{1}}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

   8. Applied egg-rr 29.0%

      \[\leadsto \color{blue}{\frac{0.012345679012345678 \cdot {x}^{-2} - 1}{\frac{-0.1111111111111111}{x} - 1}} \]

   9. Taylor expanded in x around 0 29.0%

      \[\leadsto \frac{\color{blue}{\frac{0.012345679012345678}{{x}^{2}}} - 1}{\frac{-0.1111111111111111}{x} - 1} \]
   10. Step-by-step derivation

       1. unpow2 29.0%

          \[\leadsto \frac{\frac{0.012345679012345678}{\color{blue}{x \cdot x}} - 1}{\frac{-0.1111111111111111}{x} - 1} \]

   11. Simplified 29.0%

       \[\leadsto \frac{\color{blue}{\frac{0.012345679012345678}{x \cdot x}} - 1}{\frac{-0.1111111111111111}{x} - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{-1 + \frac{-0.1111111111111111}{x \cdot \left(x \cdot 9\right)}}{\frac{-0.1111111111111111}{x} + -1}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.012345679012345678}{x \cdot x}}{\frac{-0.1111111111111111}{x} + -1}\\ \end{array} \]

Alternative 10: 65.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.012345679012345678}{x \cdot x}}{\frac{-0.1111111111111111}{x} + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e+154)
   (+ 1.0 (/ -1.0 (* x 9.0)))
   (/
    (+ -1.0 (/ 0.012345679012345678 (* x x)))
    (+ (/ -0.1111111111111111 x) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+154) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (-1.0 + (0.012345679012345678 / (x * x))) / ((-0.1111111111111111 / x) + -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 (y <= 1d+154) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = ((-1.0d0) + (0.012345679012345678d0 / (x * x))) / (((-0.1111111111111111d0) / x) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+154) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (-1.0 + (0.012345679012345678 / (x * x))) / ((-0.1111111111111111 / x) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+154:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (-1.0 + (0.012345679012345678 / (x * x))) / ((-0.1111111111111111 / x) + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+154)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(-1.0 + Float64(0.012345679012345678 / Float64(x * x))) / Float64(Float64(-0.1111111111111111 / x) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+154)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (-1.0 + (0.012345679012345678 / (x * x))) / ((-0.1111111111111111 / x) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+154], 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[(N[(-0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000004e154

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.6%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.6%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-neg-frac 99.6%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. neg-mul-1 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      7. *-commutative 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      8. associate-*r/ 99.6%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      9. fma-def 99.6%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 72.2%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} \]

      2. sqrt-unprod 42.2%

         \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} \]

      3. frac-times 42.2%

         \[\leadsto 1 + \sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}} \]

      4. metadata-eval 42.2%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 42.2%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}} \]

      6. frac-times 42.2%

         \[\leadsto 1 + \sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}} \]

      7. pow1 42.2%

         \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}} \cdot \frac{0.1111111111111111}{x}} \]

      8. pow1 42.2%

         \[\leadsto 1 + \sqrt{{\left(\frac{0.1111111111111111}{x}\right)}^{1} \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}}} \]

      9. pow-prod-up 42.2%

         \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{\left(1 + 1\right)}}} \]

      10. metadata-eval 42.2%

          \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

      11. associate-/r* 42.2%

          \[\leadsto 1 + \sqrt{{\color{blue}{\left(\frac{1}{9 \cdot x}\right)}}^{\left(1 + 1\right)}} \]

      12. *-commutative 42.2%

          \[\leadsto 1 + \sqrt{{\left(\frac{1}{\color{blue}{x \cdot 9}}\right)}^{\left(1 + 1\right)}} \]

      13. metadata-eval 42.2%

          \[\leadsto 1 + \sqrt{{\left(\frac{1}{x \cdot 9}\right)}^{\color{blue}{2}}} \]

      14. pow2 42.2%

          \[\leadsto 1 + \sqrt{\color{blue}{\frac{1}{x \cdot 9} \cdot \frac{1}{x \cdot 9}}} \]

      15. sqrt-unprod 40.3%

          \[\leadsto 1 + \color{blue}{\sqrt{\frac{1}{x \cdot 9}} \cdot \sqrt{\frac{1}{x \cdot 9}}} \]

      16. add-sqr-sqrt 40.3%

          \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot 9}} \]

      17. inv-pow 40.3%

          \[\leadsto 1 + \color{blue}{{\left(x \cdot 9\right)}^{-1}} \]

   6. Applied egg-rr 40.3%

      \[\leadsto 1 + \color{blue}{{\left(x \cdot 9\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 40.3%

         \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot 9}} \]

      2. *-commutative 40.3%

         \[\leadsto 1 + \frac{1}{\color{blue}{9 \cdot x}} \]

      3. associate-/r* 40.3%

         \[\leadsto 1 + \color{blue}{\frac{\frac{1}{9}}{x}} \]

      4. metadata-eval 40.3%

         \[\leadsto 1 + \frac{\color{blue}{0.1111111111111111}}{x} \]

      5. metadata-eval 40.3%

         \[\leadsto 1 + \frac{\color{blue}{-1 \cdot -0.1111111111111111}}{x} \]

      6. associate-*r/ 40.3%

         \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-0.1111111111111111}{x}} \]

      7. clear-num 40.3%

         \[\leadsto 1 + -1 \cdot \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}} \]

      8. un-div-inv 40.3%

         \[\leadsto 1 + \color{blue}{\frac{-1}{\frac{x}{-0.1111111111111111}}} \]

      9. clear-num 40.3%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\frac{1}{\frac{-0.1111111111111111}{x}}}} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto 1 + \frac{-1}{\frac{1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}}}} \]

      11. sqrt-unprod 53.3%

          \[\leadsto 1 + \frac{-1}{\frac{1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}}}} \]

      12. sqr-neg 53.3%

          \[\leadsto 1 + \frac{-1}{\frac{1}{\sqrt{\color{blue}{\left(-\frac{-0.1111111111111111}{x}\right) \cdot \left(-\frac{-0.1111111111111111}{x}\right)}}}} \]

      13. mul-1-neg 53.3%

          \[\leadsto 1 + \frac{-1}{\frac{1}{\sqrt{\color{blue}{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right)} \cdot \left(-\frac{-0.1111111111111111}{x}\right)}}} \]

      14. mul-1-neg 53.3%

          \[\leadsto 1 + \frac{-1}{\frac{1}{\sqrt{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right) \cdot \color{blue}{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right)}}}} \]

      15. sqrt-unprod 72.1%

          \[\leadsto 1 + \frac{-1}{\frac{1}{\color{blue}{\sqrt{-1 \cdot \frac{-0.1111111111111111}{x}} \cdot \sqrt{-1 \cdot \frac{-0.1111111111111111}{x}}}}} \]

      16. add-sqr-sqrt 72.2%

          \[\leadsto 1 + \frac{-1}{\frac{1}{\color{blue}{-1 \cdot \frac{-0.1111111111111111}{x}}}} \]

      17. associate-*r/ 72.2%

          \[\leadsto 1 + \frac{-1}{\frac{1}{\color{blue}{\frac{-1 \cdot -0.1111111111111111}{x}}}} \]

      18. metadata-eval 72.2%

          \[\leadsto 1 + \frac{-1}{\frac{1}{\frac{\color{blue}{0.1111111111111111}}{x}}} \]

      19. clear-num 72.2%

          \[\leadsto 1 + \frac{-1}{\color{blue}{\frac{x}{0.1111111111111111}}} \]

      20. div-inv 72.2%

          \[\leadsto 1 + \frac{-1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      21. metadata-eval 72.2%

          \[\leadsto 1 + \frac{-1}{x \cdot \color{blue}{9}} \]

   8. Applied egg-rr 72.2%

      \[\leadsto 1 + \color{blue}{\frac{-1}{x \cdot 9}} \]

   if 1.00000000000000004e154 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.6%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.6%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-neg-frac 99.6%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. neg-mul-1 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      7. *-commutative 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      8. associate-*r/ 99.5%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      9. fma-def 99.5%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 4.1%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} \]

      2. sqrt-unprod 0.5%

         \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} \]

      3. frac-times 0.5%

         \[\leadsto 1 + \sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}} \]

      4. metadata-eval 0.5%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 0.5%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}} \]

      6. frac-times 0.5%

         \[\leadsto 1 + \sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}} \]

      7. pow1 0.5%

         \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}} \cdot \frac{0.1111111111111111}{x}} \]

      8. pow1 0.5%

         \[\leadsto 1 + \sqrt{{\left(\frac{0.1111111111111111}{x}\right)}^{1} \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}}} \]

      9. pow-prod-up 0.5%

         \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{\left(1 + 1\right)}}} \]

      10. metadata-eval 0.5%

          \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

      11. associate-/r* 0.5%

          \[\leadsto 1 + \sqrt{{\color{blue}{\left(\frac{1}{9 \cdot x}\right)}}^{\left(1 + 1\right)}} \]

      12. *-commutative 0.5%

          \[\leadsto 1 + \sqrt{{\left(\frac{1}{\color{blue}{x \cdot 9}}\right)}^{\left(1 + 1\right)}} \]

      13. metadata-eval 0.5%

          \[\leadsto 1 + \sqrt{{\left(\frac{1}{x \cdot 9}\right)}^{\color{blue}{2}}} \]

      14. pow2 0.5%

          \[\leadsto 1 + \sqrt{\color{blue}{\frac{1}{x \cdot 9} \cdot \frac{1}{x \cdot 9}}} \]

      15. sqrt-unprod 0.6%

          \[\leadsto 1 + \color{blue}{\sqrt{\frac{1}{x \cdot 9}} \cdot \sqrt{\frac{1}{x \cdot 9}}} \]

      16. add-sqr-sqrt 0.6%

          \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot 9}} \]

      17. frac-2neg 0.6%

          \[\leadsto 1 + \color{blue}{\frac{-1}{-x \cdot 9}} \]

      18. metadata-eval 0.6%

          \[\leadsto 1 + \frac{\color{blue}{-1}}{-x \cdot 9} \]

      19. div-inv 0.6%

          \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1}{-x \cdot 9}} \]

      20. distribute-rgt-neg-in 0.6%

          \[\leadsto 1 + -1 \cdot \frac{1}{\color{blue}{x \cdot \left(-9\right)}} \]

      21. metadata-eval 0.6%

          \[\leadsto 1 + -1 \cdot \frac{1}{x \cdot \color{blue}{-9}} \]

      22. metadata-eval 0.6%

          \[\leadsto 1 + -1 \cdot \frac{1}{x \cdot \color{blue}{\frac{1}{-0.1111111111111111}}} \]

      23. div-inv 0.6%

          \[\leadsto 1 + -1 \cdot \frac{1}{\color{blue}{\frac{x}{-0.1111111111111111}}} \]

      24. clear-num 0.6%

          \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-0.1111111111111111}{x}} \]

   6. Applied egg-rr 0.6%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-0.1111111111111111}{x}} \]
   7. Step-by-step derivation

      1. +-commutative 0.6%

         \[\leadsto \color{blue}{-1 \cdot \frac{-0.1111111111111111}{x} + 1} \]

      2. flip-+ 0.5%

         \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right) \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1}} \]

      3. mul-1-neg 0.5%

         \[\leadsto \frac{\color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)} \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      4. mul-1-neg 0.5%

         \[\leadsto \frac{\left(-\frac{-0.1111111111111111}{x}\right) \cdot \color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      5. sqr-neg 0.5%

         \[\leadsto \frac{\color{blue}{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      6. div-inv 0.5%

         \[\leadsto \frac{\color{blue}{\left(-0.1111111111111111 \cdot \frac{1}{x}\right)} \cdot \frac{-0.1111111111111111}{x} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      7. div-inv 0.5%

         \[\leadsto \frac{\left(-0.1111111111111111 \cdot \frac{1}{x}\right) \cdot \color{blue}{\left(-0.1111111111111111 \cdot \frac{1}{x}\right)} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      8. swap-sqr 0.5%

         \[\leadsto \frac{\color{blue}{\left(-0.1111111111111111 \cdot -0.1111111111111111\right) \cdot \left(\frac{1}{x} \cdot \frac{1}{x}\right)} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      9. metadata-eval 0.5%

         \[\leadsto \frac{\color{blue}{0.012345679012345678} \cdot \left(\frac{1}{x} \cdot \frac{1}{x}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      10. inv-pow 0.5%

          \[\leadsto \frac{0.012345679012345678 \cdot \left(\color{blue}{{x}^{-1}} \cdot \frac{1}{x}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      11. inv-pow 0.5%

          \[\leadsto \frac{0.012345679012345678 \cdot \left({x}^{-1} \cdot \color{blue}{{x}^{-1}}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      12. pow-prod-up 0.5%

          \[\leadsto \frac{0.012345679012345678 \cdot \color{blue}{{x}^{\left(-1 + -1\right)}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      13. metadata-eval 0.5%

          \[\leadsto \frac{0.012345679012345678 \cdot {x}^{\color{blue}{-2}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      14. metadata-eval 0.5%

          \[\leadsto \frac{0.012345679012345678 \cdot {x}^{-2} - \color{blue}{1}}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

   8. Applied egg-rr 29.0%

      \[\leadsto \color{blue}{\frac{0.012345679012345678 \cdot {x}^{-2} - 1}{\frac{-0.1111111111111111}{x} - 1}} \]

   9. Taylor expanded in x around 0 29.0%

      \[\leadsto \frac{\color{blue}{\frac{0.012345679012345678}{{x}^{2}}} - 1}{\frac{-0.1111111111111111}{x} - 1} \]
   10. Step-by-step derivation

       1. unpow2 29.0%

          \[\leadsto \frac{\frac{0.012345679012345678}{\color{blue}{x \cdot x}} - 1}{\frac{-0.1111111111111111}{x} - 1} \]

   11. Simplified 29.0%

       \[\leadsto \frac{\color{blue}{\frac{0.012345679012345678}{x \cdot x}} - 1}{\frac{-0.1111111111111111}{x} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+154}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.012345679012345678}{x \cdot x}}{\frac{-0.1111111111111111}{x} + -1}\\ \end{array} \]

Alternative 11: 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 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.6%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. sub-neg 99.6%

      \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

   3. +-commutative 99.6%

      \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

   4. distribute-neg-in 99.6%

      \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

   5. distribute-neg-frac 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   6. neg-mul-1 99.6%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   7. *-commutative 99.6%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   8. associate-*r/ 99.6%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   9. fma-def 99.6%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   10. associate-/r* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   11. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   12. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   13. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   14. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   15. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in y around 0 64.2%

   \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

      \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} \]

   2. sqrt-unprod 37.3%

      \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} \]

   3. frac-times 37.3%

      \[\leadsto 1 + \sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}} \]

   4. metadata-eval 37.3%

      \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

   5. metadata-eval 37.3%

      \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}} \]

   6. frac-times 37.3%

      \[\leadsto 1 + \sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}} \]

   7. pow1 37.3%

      \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}} \cdot \frac{0.1111111111111111}{x}} \]

   8. pow1 37.3%

      \[\leadsto 1 + \sqrt{{\left(\frac{0.1111111111111111}{x}\right)}^{1} \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}}} \]

   9. pow-prod-up 37.3%

      \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{\left(1 + 1\right)}}} \]

   10. metadata-eval 37.3%

       \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

   11. associate-/r* 37.3%

       \[\leadsto 1 + \sqrt{{\color{blue}{\left(\frac{1}{9 \cdot x}\right)}}^{\left(1 + 1\right)}} \]

   12. *-commutative 37.3%

       \[\leadsto 1 + \sqrt{{\left(\frac{1}{\color{blue}{x \cdot 9}}\right)}^{\left(1 + 1\right)}} \]

   13. metadata-eval 37.3%

       \[\leadsto 1 + \sqrt{{\left(\frac{1}{x \cdot 9}\right)}^{\color{blue}{2}}} \]

   14. pow2 37.3%

       \[\leadsto 1 + \sqrt{\color{blue}{\frac{1}{x \cdot 9} \cdot \frac{1}{x \cdot 9}}} \]

   15. sqrt-unprod 35.6%

       \[\leadsto 1 + \color{blue}{\sqrt{\frac{1}{x \cdot 9}} \cdot \sqrt{\frac{1}{x \cdot 9}}} \]

   16. add-sqr-sqrt 35.6%

       \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot 9}} \]

   17. inv-pow 35.6%

       \[\leadsto 1 + \color{blue}{{\left(x \cdot 9\right)}^{-1}} \]

6. Applied egg-rr 35.6%

   \[\leadsto 1 + \color{blue}{{\left(x \cdot 9\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 35.6%

      \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot 9}} \]

   2. *-commutative 35.6%

      \[\leadsto 1 + \frac{1}{\color{blue}{9 \cdot x}} \]

   3. associate-/r* 35.6%

      \[\leadsto 1 + \color{blue}{\frac{\frac{1}{9}}{x}} \]

   4. metadata-eval 35.6%

      \[\leadsto 1 + \frac{\color{blue}{0.1111111111111111}}{x} \]

   5. metadata-eval 35.6%

      \[\leadsto 1 + \frac{\color{blue}{-1 \cdot -0.1111111111111111}}{x} \]

   6. associate-*r/ 35.6%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-0.1111111111111111}{x}} \]

   7. clear-num 35.6%

      \[\leadsto 1 + -1 \cdot \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}} \]

   8. un-div-inv 35.6%

      \[\leadsto 1 + \color{blue}{\frac{-1}{\frac{x}{-0.1111111111111111}}} \]

   9. clear-num 35.6%

      \[\leadsto 1 + \frac{-1}{\color{blue}{\frac{1}{\frac{-0.1111111111111111}{x}}}} \]

   10. add-sqr-sqrt 0.0%

       \[\leadsto 1 + \frac{-1}{\frac{1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}}}} \]

   11. sqrt-unprod 50.5%

       \[\leadsto 1 + \frac{-1}{\frac{1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}}}} \]

   12. sqr-neg 50.5%

       \[\leadsto 1 + \frac{-1}{\frac{1}{\sqrt{\color{blue}{\left(-\frac{-0.1111111111111111}{x}\right) \cdot \left(-\frac{-0.1111111111111111}{x}\right)}}}} \]

   13. mul-1-neg 50.5%

       \[\leadsto 1 + \frac{-1}{\frac{1}{\sqrt{\color{blue}{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right)} \cdot \left(-\frac{-0.1111111111111111}{x}\right)}}} \]

   14. mul-1-neg 50.5%

       \[\leadsto 1 + \frac{-1}{\frac{1}{\sqrt{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right) \cdot \color{blue}{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right)}}}} \]

   15. sqrt-unprod 64.1%

       \[\leadsto 1 + \frac{-1}{\frac{1}{\color{blue}{\sqrt{-1 \cdot \frac{-0.1111111111111111}{x}} \cdot \sqrt{-1 \cdot \frac{-0.1111111111111111}{x}}}}} \]

   16. add-sqr-sqrt 64.2%

       \[\leadsto 1 + \frac{-1}{\frac{1}{\color{blue}{-1 \cdot \frac{-0.1111111111111111}{x}}}} \]

   17. associate-*r/ 64.2%

       \[\leadsto 1 + \frac{-1}{\frac{1}{\color{blue}{\frac{-1 \cdot -0.1111111111111111}{x}}}} \]

   18. metadata-eval 64.2%

       \[\leadsto 1 + \frac{-1}{\frac{1}{\frac{\color{blue}{0.1111111111111111}}{x}}} \]

   19. clear-num 64.2%

       \[\leadsto 1 + \frac{-1}{\color{blue}{\frac{x}{0.1111111111111111}}} \]

   20. div-inv 64.3%

       \[\leadsto 1 + \frac{-1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

   21. metadata-eval 64.3%

       \[\leadsto 1 + \frac{-1}{x \cdot \color{blue}{9}} \]

8. Applied egg-rr 64.3%

   \[\leadsto 1 + \color{blue}{\frac{-1}{x \cdot 9}} \]

9. Final simplification 64.3%

   \[\leadsto 1 + \frac{-1}{x \cdot 9} \]

Alternative 12: 61.7% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.5e-7) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.5e-7) {
		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 <= 4.5d-7) 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 <= 4.5e-7) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.5e-7:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.5e-7)
		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 <= 4.5e-7)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.5e-7], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.4999999999999998e-7

   1. Initial program 99.5%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.5%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.5%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.5%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.5%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-neg-frac 99.5%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. neg-mul-1 99.5%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      7. *-commutative 99.5%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      8. associate-*r/ 99.5%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      9. fma-def 99.5%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 60.1%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} \]

      2. sqrt-unprod 5.2%

         \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} \]

      3. frac-times 5.2%

         \[\leadsto 1 + \sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}} \]

      4. metadata-eval 5.2%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 5.2%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}} \]

      6. frac-times 5.2%

         \[\leadsto 1 + \sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}} \]

      7. pow1 5.2%

         \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}} \cdot \frac{0.1111111111111111}{x}} \]

      8. pow1 5.2%

         \[\leadsto 1 + \sqrt{{\left(\frac{0.1111111111111111}{x}\right)}^{1} \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}}} \]

      9. pow-prod-up 5.2%

         \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{\left(1 + 1\right)}}} \]

      10. metadata-eval 5.2%

          \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

      11. associate-/r* 5.2%

          \[\leadsto 1 + \sqrt{{\color{blue}{\left(\frac{1}{9 \cdot x}\right)}}^{\left(1 + 1\right)}} \]

      12. *-commutative 5.2%

          \[\leadsto 1 + \sqrt{{\left(\frac{1}{\color{blue}{x \cdot 9}}\right)}^{\left(1 + 1\right)}} \]

      13. metadata-eval 5.2%

          \[\leadsto 1 + \sqrt{{\left(\frac{1}{x \cdot 9}\right)}^{\color{blue}{2}}} \]

      14. pow2 5.2%

          \[\leadsto 1 + \sqrt{\color{blue}{\frac{1}{x \cdot 9} \cdot \frac{1}{x \cdot 9}}} \]

      15. sqrt-unprod 1.6%

          \[\leadsto 1 + \color{blue}{\sqrt{\frac{1}{x \cdot 9}} \cdot \sqrt{\frac{1}{x \cdot 9}}} \]

      16. add-sqr-sqrt 1.6%

          \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot 9}} \]

      17. frac-2neg 1.6%

          \[\leadsto 1 + \color{blue}{\frac{-1}{-x \cdot 9}} \]

      18. metadata-eval 1.6%

          \[\leadsto 1 + \frac{\color{blue}{-1}}{-x \cdot 9} \]

      19. div-inv 1.6%

          \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1}{-x \cdot 9}} \]

      20. distribute-rgt-neg-in 1.6%

          \[\leadsto 1 + -1 \cdot \frac{1}{\color{blue}{x \cdot \left(-9\right)}} \]

      21. metadata-eval 1.6%

          \[\leadsto 1 + -1 \cdot \frac{1}{x \cdot \color{blue}{-9}} \]

      22. metadata-eval 1.6%

          \[\leadsto 1 + -1 \cdot \frac{1}{x \cdot \color{blue}{\frac{1}{-0.1111111111111111}}} \]

      23. div-inv 1.6%

          \[\leadsto 1 + -1 \cdot \frac{1}{\color{blue}{\frac{x}{-0.1111111111111111}}} \]

      24. clear-num 1.6%

          \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-0.1111111111111111}{x}} \]

   6. Applied egg-rr 1.6%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-0.1111111111111111}{x}} \]
   7. Step-by-step derivation

      1. +-commutative 1.6%

         \[\leadsto \color{blue}{-1 \cdot \frac{-0.1111111111111111}{x} + 1} \]

      2. flip-+ 5.2%

         \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right) \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1}} \]

      3. mul-1-neg 5.2%

         \[\leadsto \frac{\color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)} \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      4. mul-1-neg 5.2%

         \[\leadsto \frac{\left(-\frac{-0.1111111111111111}{x}\right) \cdot \color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      5. sqr-neg 5.2%

         \[\leadsto \frac{\color{blue}{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      6. div-inv 5.2%

         \[\leadsto \frac{\color{blue}{\left(-0.1111111111111111 \cdot \frac{1}{x}\right)} \cdot \frac{-0.1111111111111111}{x} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      7. div-inv 5.2%

         \[\leadsto \frac{\left(-0.1111111111111111 \cdot \frac{1}{x}\right) \cdot \color{blue}{\left(-0.1111111111111111 \cdot \frac{1}{x}\right)} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      8. swap-sqr 5.2%

         \[\leadsto \frac{\color{blue}{\left(-0.1111111111111111 \cdot -0.1111111111111111\right) \cdot \left(\frac{1}{x} \cdot \frac{1}{x}\right)} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      9. metadata-eval 5.2%

         \[\leadsto \frac{\color{blue}{0.012345679012345678} \cdot \left(\frac{1}{x} \cdot \frac{1}{x}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      10. inv-pow 5.2%

          \[\leadsto \frac{0.012345679012345678 \cdot \left(\color{blue}{{x}^{-1}} \cdot \frac{1}{x}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      11. inv-pow 5.2%

          \[\leadsto \frac{0.012345679012345678 \cdot \left({x}^{-1} \cdot \color{blue}{{x}^{-1}}\right) - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      12. pow-prod-up 5.2%

          \[\leadsto \frac{0.012345679012345678 \cdot \color{blue}{{x}^{\left(-1 + -1\right)}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      13. metadata-eval 5.2%

          \[\leadsto \frac{0.012345679012345678 \cdot {x}^{\color{blue}{-2}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      14. metadata-eval 5.2%

          \[\leadsto \frac{0.012345679012345678 \cdot {x}^{-2} - \color{blue}{1}}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

   8. Applied egg-rr 31.7%

      \[\leadsto \color{blue}{\frac{0.012345679012345678 \cdot {x}^{-2} - 1}{\frac{-0.1111111111111111}{x} - 1}} \]

   9. Taylor expanded in x around 0 59.4%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} \]

   if 4.4999999999999998e-7 < 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. sub-neg 99.8%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.8%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.8%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-neg-frac 99.8%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. neg-mul-1 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      7. *-commutative 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      8. associate-*r/ 99.8%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      9. fma-def 99.8%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 68.1%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} \]

      2. sqrt-unprod 67.5%

         \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} \]

      3. frac-times 67.5%

         \[\leadsto 1 + \sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}} \]

      4. metadata-eval 67.5%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

      5. metadata-eval 67.5%

         \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}} \]

      6. frac-times 67.5%

         \[\leadsto 1 + \sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}} \]

      7. pow1 67.5%

         \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}} \cdot \frac{0.1111111111111111}{x}} \]

      8. pow1 67.5%

         \[\leadsto 1 + \sqrt{{\left(\frac{0.1111111111111111}{x}\right)}^{1} \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}}} \]

      9. pow-prod-up 67.5%

         \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{\left(1 + 1\right)}}} \]

      10. metadata-eval 67.5%

          \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

      11. associate-/r* 67.5%

          \[\leadsto 1 + \sqrt{{\color{blue}{\left(\frac{1}{9 \cdot x}\right)}}^{\left(1 + 1\right)}} \]

      12. *-commutative 67.5%

          \[\leadsto 1 + \sqrt{{\left(\frac{1}{\color{blue}{x \cdot 9}}\right)}^{\left(1 + 1\right)}} \]

      13. metadata-eval 67.5%

          \[\leadsto 1 + \sqrt{{\left(\frac{1}{x \cdot 9}\right)}^{\color{blue}{2}}} \]

      14. pow2 67.5%

          \[\leadsto 1 + \sqrt{\color{blue}{\frac{1}{x \cdot 9} \cdot \frac{1}{x \cdot 9}}} \]

      15. sqrt-unprod 67.5%

          \[\leadsto 1 + \color{blue}{\sqrt{\frac{1}{x \cdot 9}} \cdot \sqrt{\frac{1}{x \cdot 9}}} \]

      16. add-sqr-sqrt 67.5%

          \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot 9}} \]

      17. frac-2neg 67.5%

          \[\leadsto 1 + \color{blue}{\frac{-1}{-x \cdot 9}} \]

      18. metadata-eval 67.5%

          \[\leadsto 1 + \frac{\color{blue}{-1}}{-x \cdot 9} \]

      19. div-inv 67.5%

          \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1}{-x \cdot 9}} \]

      20. distribute-rgt-neg-in 67.5%

          \[\leadsto 1 + -1 \cdot \frac{1}{\color{blue}{x \cdot \left(-9\right)}} \]

      21. metadata-eval 67.5%

          \[\leadsto 1 + -1 \cdot \frac{1}{x \cdot \color{blue}{-9}} \]

      22. metadata-eval 67.5%

          \[\leadsto 1 + -1 \cdot \frac{1}{x \cdot \color{blue}{\frac{1}{-0.1111111111111111}}} \]

      23. div-inv 67.5%

          \[\leadsto 1 + -1 \cdot \frac{1}{\color{blue}{\frac{x}{-0.1111111111111111}}} \]

      24. clear-num 67.5%

          \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-0.1111111111111111}{x}} \]

   6. Applied egg-rr 67.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-0.1111111111111111}{x}} \]

   7. Taylor expanded in x around inf 67.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 62.7% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ -0.1111111111111111 x)))

C

double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-0.1111111111111111d0) / x)
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

function code(x, y)
	return Float64(1.0 + Float64(-0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-0.1111111111111111 / x);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.6%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. sub-neg 99.6%

      \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

   3. +-commutative 99.6%

      \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

   4. distribute-neg-in 99.6%

      \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

   5. distribute-neg-frac 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   6. neg-mul-1 99.6%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   7. *-commutative 99.6%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   8. associate-*r/ 99.6%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   9. fma-def 99.6%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   10. associate-/r* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   11. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   12. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   13. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   14. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   15. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in y around 0 64.2%

   \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]

5. Final simplification 64.2%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]

Alternative 14: 32.2% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.6%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. sub-neg 99.6%

      \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

   3. +-commutative 99.6%

      \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

   4. distribute-neg-in 99.6%

      \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

   5. distribute-neg-frac 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   6. neg-mul-1 99.6%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   7. *-commutative 99.6%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   8. associate-*r/ 99.6%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   9. fma-def 99.6%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   10. associate-/r* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   11. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   12. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   13. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   14. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   15. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in y around 0 64.2%

   \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

      \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} \]

   2. sqrt-unprod 37.3%

      \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} \]

   3. frac-times 37.3%

      \[\leadsto 1 + \sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}} \]

   4. metadata-eval 37.3%

      \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

   5. metadata-eval 37.3%

      \[\leadsto 1 + \sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}} \]

   6. frac-times 37.3%

      \[\leadsto 1 + \sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}} \]

   7. pow1 37.3%

      \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}} \cdot \frac{0.1111111111111111}{x}} \]

   8. pow1 37.3%

      \[\leadsto 1 + \sqrt{{\left(\frac{0.1111111111111111}{x}\right)}^{1} \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1}}} \]

   9. pow-prod-up 37.3%

      \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{\left(1 + 1\right)}}} \]

   10. metadata-eval 37.3%

       \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

   11. associate-/r* 37.3%

       \[\leadsto 1 + \sqrt{{\color{blue}{\left(\frac{1}{9 \cdot x}\right)}}^{\left(1 + 1\right)}} \]

   12. *-commutative 37.3%

       \[\leadsto 1 + \sqrt{{\left(\frac{1}{\color{blue}{x \cdot 9}}\right)}^{\left(1 + 1\right)}} \]

   13. metadata-eval 37.3%

       \[\leadsto 1 + \sqrt{{\left(\frac{1}{x \cdot 9}\right)}^{\color{blue}{2}}} \]

   14. pow2 37.3%

       \[\leadsto 1 + \sqrt{\color{blue}{\frac{1}{x \cdot 9} \cdot \frac{1}{x \cdot 9}}} \]

   15. sqrt-unprod 35.6%

       \[\leadsto 1 + \color{blue}{\sqrt{\frac{1}{x \cdot 9}} \cdot \sqrt{\frac{1}{x \cdot 9}}} \]

   16. add-sqr-sqrt 35.6%

       \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot 9}} \]

   17. frac-2neg 35.6%

       \[\leadsto 1 + \color{blue}{\frac{-1}{-x \cdot 9}} \]

   18. metadata-eval 35.6%

       \[\leadsto 1 + \frac{\color{blue}{-1}}{-x \cdot 9} \]

   19. div-inv 35.6%

       \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1}{-x \cdot 9}} \]

   20. distribute-rgt-neg-in 35.6%

       \[\leadsto 1 + -1 \cdot \frac{1}{\color{blue}{x \cdot \left(-9\right)}} \]

   21. metadata-eval 35.6%

       \[\leadsto 1 + -1 \cdot \frac{1}{x \cdot \color{blue}{-9}} \]

   22. metadata-eval 35.6%

       \[\leadsto 1 + -1 \cdot \frac{1}{x \cdot \color{blue}{\frac{1}{-0.1111111111111111}}} \]

   23. div-inv 35.6%

       \[\leadsto 1 + -1 \cdot \frac{1}{\color{blue}{\frac{x}{-0.1111111111111111}}} \]

   24. clear-num 35.6%

       \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-0.1111111111111111}{x}} \]

6. Applied egg-rr 35.6%

   \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-0.1111111111111111}{x}} \]

7. Taylor expanded in x around inf 35.5%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 35.5%

   \[\leadsto 1 \]

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 2023230 
(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)))))