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

Percentage Accurate: 99.7% → 99.6%

Time: 14.1s

Alternatives: 17

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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) - y \cdot \left(0.3333333333333333 \cdot {x}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (-
  (- 1.0 (/ 0.1111111111111111 x))
  (* y (* 0.3333333333333333 (pow x -0.5)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y * (0.3333333333333333 * pow(x, -0.5)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) - (y * (0.3333333333333333d0 * (x ** (-0.5d0))))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y * (0.3333333333333333 * Math.pow(x, -0.5)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) - (y * (0.3333333333333333 * math.pow(x, -0.5)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) - Float64(y * Float64(0.3333333333333333 * (x ^ -0.5))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) - (y * (0.3333333333333333 * (x ^ -0.5)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y * N[(0.3333333333333333 * N[Power[x, -0.5], $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. Add Preprocessing

3. Taylor expanded in x around 0 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.7%

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

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

   5. pow-flip 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

   6. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

   7. div-inv 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{-0.5} \cdot \color{blue}{\left(y \cdot \frac{1}{3}\right)} \]

   8. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{-0.5} \cdot \left(y \cdot \color{blue}{0.3333333333333333}\right) \]

5. Applied egg-rr 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\left(y \cdot 0.3333333333333333\right) \cdot {x}^{-0.5}} \]

   2. associate-*l* 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{y \cdot \left(0.3333333333333333 \cdot {x}^{-0.5}\right)} \]

7. Simplified 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{y \cdot \left(0.3333333333333333 \cdot {x}^{-0.5}\right)} \]

8. Final simplification 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - y \cdot \left(0.3333333333333333 \cdot {x}^{-0.5}\right) \]
9. Add Preprocessing

Alternative 2: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+65}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+55}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+65)
   (* -0.3333333333333333 (* y (pow x -0.5)))
   (if (<= y 6.8e+55)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (* -0.3333333333333333 (/ 1.0 (/ (sqrt x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+65) {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	} else if (y <= 6.8e+55) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = -0.3333333333333333 * (1.0 / (sqrt(x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.8d+65)) then
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    else if (y <= 6.8d+55) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = (-0.3333333333333333d0) * (1.0d0 / (sqrt(x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+65) {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	} else if (y <= 6.8e+55) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = -0.3333333333333333 * (1.0 / (Math.sqrt(x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+65:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	elif y <= 6.8e+55:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = -0.3333333333333333 * (1.0 / (math.sqrt(x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+65)
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	elseif (y <= 6.8e+55)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(-0.3333333333333333 * Float64(1.0 / Float64(sqrt(x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.8e+65)
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	elseif (y <= 6.8e+55)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = -0.3333333333333333 * (1.0 / (sqrt(x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+65], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+55], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8000000000000003e65

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 90.4%

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

      1. pow1/2 90.4%

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

      2. add0 90.4%

         \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\left({\left(\frac{1}{x}\right)}^{0.5} + 0\right)} \cdot y\right) \]

      3. inv-pow 90.4%

         \[\leadsto -0.3333333333333333 \cdot \left(\left({\color{blue}{\left({x}^{-1}\right)}}^{0.5} + 0\right) \cdot y\right) \]

      4. pow-pow 90.5%

         \[\leadsto -0.3333333333333333 \cdot \left(\left(\color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} + 0\right) \cdot y\right) \]

      5. metadata-eval 90.5%

         \[\leadsto -0.3333333333333333 \cdot \left(\left({x}^{\color{blue}{-0.5}} + 0\right) \cdot y\right) \]

   6. Applied egg-rr 90.5%

      \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\left({x}^{-0.5} + 0\right)} \cdot y\right) \]
   7. Step-by-step derivation

      1. add0 90.5%

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

   8. Simplified 90.5%

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

   if -4.8000000000000003e65 < y < 6.7999999999999996e55

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 97.2%

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

      1. div-inv 97.2%

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

      2. add0 97.2%

         \[\leadsto 1 - \color{blue}{\left(\frac{0.1111111111111111}{x} + 0\right)} \]

      3. flip-+ 71.6%

         \[\leadsto 1 - \color{blue}{\frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0}} \]

      4. div-inv 71.6%

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

      5. *-commutative 71.6%

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

      6. div-inv 71.5%

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

      7. *-commutative 71.5%

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

      8. swap-sqr 71.5%

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

      9. inv-pow 71.5%

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

      10. inv-pow 71.5%

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

      11. pow-prod-up 71.6%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-1 + -1\right)}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      12. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-2}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      13. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot \color{blue}{0.012345679012345678} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      14. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - \color{blue}{0}}{\frac{0.1111111111111111}{x} - 0} \]

   7. Applied egg-rr 71.6%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\frac{0.1111111111111111}{x} - 0}} \]
   8. Step-by-step derivation

      1. --rgt-identity 71.6%

         \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

      2. --rgt-identity 71.6%

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

   9. Simplified 71.6%

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

       1. add0 71.6%

          \[\leadsto 1 - \color{blue}{\left(\frac{{x}^{-2} \cdot 0.012345679012345678}{\frac{0.1111111111111111}{x}} + 0\right)} \]

       2. associate-/r/ 71.6%

          \[\leadsto 1 - \left(\color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111} \cdot x} + 0\right) \]

       3. *-commutative 71.6%

          \[\leadsto 1 - \left(\color{blue}{x \cdot \frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111}} + 0\right) \]

       4. associate-/l* 71.6%

          \[\leadsto 1 - \left(x \cdot \color{blue}{\frac{{x}^{-2}}{\frac{0.1111111111111111}{0.012345679012345678}}} + 0\right) \]

       5. metadata-eval 71.6%

          \[\leadsto 1 - \left(x \cdot \frac{{x}^{-2}}{\color{blue}{9}} + 0\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto 1 - \color{blue}{\left(x \cdot \frac{{x}^{-2}}{9} + 0\right)} \]
   12. Step-by-step derivation

       1. add0 71.6%

          \[\leadsto 1 - \color{blue}{x \cdot \frac{{x}^{-2}}{9}} \]

       2. *-commutative 71.6%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2}}{9} \cdot x} \]

       3. associate-*l/ 71.5%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot x}{9}} \]

       4. pow-plus 97.3%

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

       5. metadata-eval 97.3%

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

       6. unpow-1 97.3%

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

   13. Simplified 97.3%

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

   if 6.7999999999999996e55 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 93.0%

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

      1. sqrt-div 93.0%

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

      2. metadata-eval 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. associate-*l/ 93.0%

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

      2. associate-/l* 93.0%

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

   8. Applied egg-rr 93.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+65}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+55}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{\sqrt{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+65}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+62}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.5e+65)
   (* -0.3333333333333333 (* y (pow x -0.5)))
   (if (<= y 3e+62)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (* y (* -0.3333333333333333 (sqrt (/ 1.0 x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.5e+65) {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	} else if (y <= 3e+62) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.5d+65)) then
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    else if (y <= 3d+62) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = y * ((-0.3333333333333333d0) * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.5e+65) {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	} else if (y <= 3e+62) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = y * (-0.3333333333333333 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.5e+65:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	elif y <= 3e+62:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = y * (-0.3333333333333333 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.5e+65)
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	elseif (y <= 3e+62)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.5e+65)
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	elseif (y <= 3e+62)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.5e+65], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+62], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4999999999999996e65

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 90.4%

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

      1. pow1/2 90.4%

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

      2. add0 90.4%

         \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\left({\left(\frac{1}{x}\right)}^{0.5} + 0\right)} \cdot y\right) \]

      3. inv-pow 90.4%

         \[\leadsto -0.3333333333333333 \cdot \left(\left({\color{blue}{\left({x}^{-1}\right)}}^{0.5} + 0\right) \cdot y\right) \]

      4. pow-pow 90.5%

         \[\leadsto -0.3333333333333333 \cdot \left(\left(\color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} + 0\right) \cdot y\right) \]

      5. metadata-eval 90.5%

         \[\leadsto -0.3333333333333333 \cdot \left(\left({x}^{\color{blue}{-0.5}} + 0\right) \cdot y\right) \]

   6. Applied egg-rr 90.5%

      \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\left({x}^{-0.5} + 0\right)} \cdot y\right) \]
   7. Step-by-step derivation

      1. add0 90.5%

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

   8. Simplified 90.5%

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

   if -5.4999999999999996e65 < y < 3e62

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 97.2%

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

      1. div-inv 97.2%

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

      2. add0 97.2%

         \[\leadsto 1 - \color{blue}{\left(\frac{0.1111111111111111}{x} + 0\right)} \]

      3. flip-+ 71.6%

         \[\leadsto 1 - \color{blue}{\frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0}} \]

      4. div-inv 71.6%

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

      5. *-commutative 71.6%

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

      6. div-inv 71.5%

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

      7. *-commutative 71.5%

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

      8. swap-sqr 71.5%

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

      9. inv-pow 71.5%

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

      10. inv-pow 71.5%

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

      11. pow-prod-up 71.6%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-1 + -1\right)}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      12. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-2}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      13. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot \color{blue}{0.012345679012345678} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      14. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - \color{blue}{0}}{\frac{0.1111111111111111}{x} - 0} \]

   7. Applied egg-rr 71.6%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\frac{0.1111111111111111}{x} - 0}} \]
   8. Step-by-step derivation

      1. --rgt-identity 71.6%

         \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

      2. --rgt-identity 71.6%

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

   9. Simplified 71.6%

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

       1. add0 71.6%

          \[\leadsto 1 - \color{blue}{\left(\frac{{x}^{-2} \cdot 0.012345679012345678}{\frac{0.1111111111111111}{x}} + 0\right)} \]

       2. associate-/r/ 71.6%

          \[\leadsto 1 - \left(\color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111} \cdot x} + 0\right) \]

       3. *-commutative 71.6%

          \[\leadsto 1 - \left(\color{blue}{x \cdot \frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111}} + 0\right) \]

       4. associate-/l* 71.6%

          \[\leadsto 1 - \left(x \cdot \color{blue}{\frac{{x}^{-2}}{\frac{0.1111111111111111}{0.012345679012345678}}} + 0\right) \]

       5. metadata-eval 71.6%

          \[\leadsto 1 - \left(x \cdot \frac{{x}^{-2}}{\color{blue}{9}} + 0\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto 1 - \color{blue}{\left(x \cdot \frac{{x}^{-2}}{9} + 0\right)} \]
   12. Step-by-step derivation

       1. add0 71.6%

          \[\leadsto 1 - \color{blue}{x \cdot \frac{{x}^{-2}}{9}} \]

       2. *-commutative 71.6%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2}}{9} \cdot x} \]

       3. associate-*l/ 71.5%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot x}{9}} \]

       4. pow-plus 97.3%

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

       5. metadata-eval 97.3%

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

       6. unpow-1 97.3%

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

   13. Simplified 97.3%

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

   if 3e62 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.5%

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

      4. pow1/2 99.5%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

      5. pow-flip 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

      6. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

      7. div-inv 99.5%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{-0.5} \cdot \color{blue}{\left(y \cdot \frac{1}{3}\right)} \]

      8. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{-0.5} \cdot \left(y \cdot \color{blue}{0.3333333333333333}\right) \]

   5. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)} \]
   6. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\left(y \cdot 0.3333333333333333\right) \cdot {x}^{-0.5}} \]

      2. associate-*l* 99.5%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{y \cdot \left(0.3333333333333333 \cdot {x}^{-0.5}\right)} \]

   7. Simplified 99.5%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{y \cdot \left(0.3333333333333333 \cdot {x}^{-0.5}\right)} \]

   8. Taylor expanded in y around inf 93.0%

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

      1. associate-*r* 93.0%

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

   10. Simplified 93.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+65}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+62}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+65} \lor \neg \left(y \leq 1.85 \cdot 10^{+62}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.5e+65) (not (<= y 1.85e+62)))
   (* -0.3333333333333333 (* y (pow x -0.5)))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.5e+65) || !(y <= 1.85e+62)) {
		tmp = -0.3333333333333333 * (y * 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 <= (-5.5d+65)) .or. (.not. (y <= 1.85d+62))) then
        tmp = (-0.3333333333333333d0) * (y * (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 <= -5.5e+65) || !(y <= 1.85e+62)) {
		tmp = -0.3333333333333333 * (y * 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 <= -5.5e+65) or not (y <= 1.85e+62):
		tmp = -0.3333333333333333 * (y * 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 <= -5.5e+65) || !(y <= 1.85e+62))
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.5e+65) || ~((y <= 1.85e+62)))
		tmp = -0.3333333333333333 * (y * (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, -5.5e+65], N[Not[LessEqual[y, 1.85e+62]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4999999999999996e65 or 1.85000000000000007e62 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 91.6%

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

      1. pow1/2 91.6%

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

      2. add0 91.6%

         \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\left({\left(\frac{1}{x}\right)}^{0.5} + 0\right)} \cdot y\right) \]

      3. inv-pow 91.6%

         \[\leadsto -0.3333333333333333 \cdot \left(\left({\color{blue}{\left({x}^{-1}\right)}}^{0.5} + 0\right) \cdot y\right) \]

      4. pow-pow 91.7%

         \[\leadsto -0.3333333333333333 \cdot \left(\left(\color{blue}{{x}^{\left(-1 \cdot 0.5\right)}} + 0\right) \cdot y\right) \]

      5. metadata-eval 91.7%

         \[\leadsto -0.3333333333333333 \cdot \left(\left({x}^{\color{blue}{-0.5}} + 0\right) \cdot y\right) \]

   6. Applied egg-rr 91.7%

      \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\left({x}^{-0.5} + 0\right)} \cdot y\right) \]
   7. Step-by-step derivation

      1. add0 91.7%

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

   8. Simplified 91.7%

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

   if -5.4999999999999996e65 < y < 1.85000000000000007e62

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 97.2%

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

      1. div-inv 97.2%

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

      2. add0 97.2%

         \[\leadsto 1 - \color{blue}{\left(\frac{0.1111111111111111}{x} + 0\right)} \]

      3. flip-+ 71.6%

         \[\leadsto 1 - \color{blue}{\frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0}} \]

      4. div-inv 71.6%

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

      5. *-commutative 71.6%

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

      6. div-inv 71.5%

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

      7. *-commutative 71.5%

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

      8. swap-sqr 71.5%

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

      9. inv-pow 71.5%

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

      10. inv-pow 71.5%

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

      11. pow-prod-up 71.6%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-1 + -1\right)}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      12. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-2}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      13. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot \color{blue}{0.012345679012345678} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      14. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - \color{blue}{0}}{\frac{0.1111111111111111}{x} - 0} \]

   7. Applied egg-rr 71.6%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\frac{0.1111111111111111}{x} - 0}} \]
   8. Step-by-step derivation

      1. --rgt-identity 71.6%

         \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

      2. --rgt-identity 71.6%

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

   9. Simplified 71.6%

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

       1. add0 71.6%

          \[\leadsto 1 - \color{blue}{\left(\frac{{x}^{-2} \cdot 0.012345679012345678}{\frac{0.1111111111111111}{x}} + 0\right)} \]

       2. associate-/r/ 71.6%

          \[\leadsto 1 - \left(\color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111} \cdot x} + 0\right) \]

       3. *-commutative 71.6%

          \[\leadsto 1 - \left(\color{blue}{x \cdot \frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111}} + 0\right) \]

       4. associate-/l* 71.6%

          \[\leadsto 1 - \left(x \cdot \color{blue}{\frac{{x}^{-2}}{\frac{0.1111111111111111}{0.012345679012345678}}} + 0\right) \]

       5. metadata-eval 71.6%

          \[\leadsto 1 - \left(x \cdot \frac{{x}^{-2}}{\color{blue}{9}} + 0\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto 1 - \color{blue}{\left(x \cdot \frac{{x}^{-2}}{9} + 0\right)} \]
   12. Step-by-step derivation

       1. add0 71.6%

          \[\leadsto 1 - \color{blue}{x \cdot \frac{{x}^{-2}}{9}} \]

       2. *-commutative 71.6%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2}}{9} \cdot x} \]

       3. associate-*l/ 71.5%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot x}{9}} \]

       4. pow-plus 97.3%

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

       5. metadata-eval 97.3%

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

       6. unpow-1 97.3%

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

   13. Simplified 97.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+65} \lor \neg \left(y \leq 1.85 \cdot 10^{+62}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+65} \lor \neg \left(y \leq 2.9 \cdot 10^{+62}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.5e+65) (not (<= y 2.9e+62)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.5e+65) || !(y <= 2.9e+62)) {
		tmp = -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 <= (-5.5d+65)) .or. (.not. (y <= 2.9d+62))) then
        tmp = (-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 <= -5.5e+65) || !(y <= 2.9e+62)) {
		tmp = -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 <= -5.5e+65) or not (y <= 2.9e+62):
		tmp = -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 <= -5.5e+65) || !(y <= 2.9e+62))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.5e+65) || ~((y <= 2.9e+62)))
		tmp = -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, -5.5e+65], N[Not[LessEqual[y, 2.9e+62]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4999999999999996e65 or 2.89999999999999984e62 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 91.6%

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

      1. add0 91.6%

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

      2. *-commutative 91.6%

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

      3. sqrt-div 91.6%

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

      4. metadata-eval 91.6%

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

      5. un-div-inv 91.6%

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

   6. Applied egg-rr 91.6%

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

      1. add0 91.6%

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

   8. Simplified 91.6%

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

   if -5.4999999999999996e65 < y < 2.89999999999999984e62

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 97.2%

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

      1. div-inv 97.2%

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

      2. add0 97.2%

         \[\leadsto 1 - \color{blue}{\left(\frac{0.1111111111111111}{x} + 0\right)} \]

      3. flip-+ 71.6%

         \[\leadsto 1 - \color{blue}{\frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0}} \]

      4. div-inv 71.6%

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

      5. *-commutative 71.6%

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

      6. div-inv 71.5%

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

      7. *-commutative 71.5%

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

      8. swap-sqr 71.5%

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

      9. inv-pow 71.5%

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

      10. inv-pow 71.5%

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

      11. pow-prod-up 71.6%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-1 + -1\right)}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      12. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-2}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      13. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot \color{blue}{0.012345679012345678} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      14. metadata-eval 71.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - \color{blue}{0}}{\frac{0.1111111111111111}{x} - 0} \]

   7. Applied egg-rr 71.6%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\frac{0.1111111111111111}{x} - 0}} \]
   8. Step-by-step derivation

      1. --rgt-identity 71.6%

         \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

      2. --rgt-identity 71.6%

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

   9. Simplified 71.6%

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

       1. add0 71.6%

          \[\leadsto 1 - \color{blue}{\left(\frac{{x}^{-2} \cdot 0.012345679012345678}{\frac{0.1111111111111111}{x}} + 0\right)} \]

       2. associate-/r/ 71.6%

          \[\leadsto 1 - \left(\color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111} \cdot x} + 0\right) \]

       3. *-commutative 71.6%

          \[\leadsto 1 - \left(\color{blue}{x \cdot \frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111}} + 0\right) \]

       4. associate-/l* 71.6%

          \[\leadsto 1 - \left(x \cdot \color{blue}{\frac{{x}^{-2}}{\frac{0.1111111111111111}{0.012345679012345678}}} + 0\right) \]

       5. metadata-eval 71.6%

          \[\leadsto 1 - \left(x \cdot \frac{{x}^{-2}}{\color{blue}{9}} + 0\right) \]

   11. Applied egg-rr 71.6%

       \[\leadsto 1 - \color{blue}{\left(x \cdot \frac{{x}^{-2}}{9} + 0\right)} \]
   12. Step-by-step derivation

       1. add0 71.6%

          \[\leadsto 1 - \color{blue}{x \cdot \frac{{x}^{-2}}{9}} \]

       2. *-commutative 71.6%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2}}{9} \cdot x} \]

       3. associate-*l/ 71.5%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot x}{9}} \]

       4. pow-plus 97.3%

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

       5. metadata-eval 97.3%

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

       6. unpow-1 97.3%

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

   13. Simplified 97.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+65} \lor \neg \left(y \leq 2.9 \cdot 10^{+62}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 99.6%

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

   1. associate-*r* 99.6%

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

7. Simplified 99.6%

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

   1. add0 99.6%

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

   2. +-commutative 99.6%

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

   3. sqrt-div 99.6%

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

   4. metadata-eval 99.6%

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

   5. un-div-inv 99.7%

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

9. Applied egg-rr 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\left(0 + \frac{-0.3333333333333333}{\sqrt{x}}\right)} \cdot y \]
10. Step-by-step derivation

    1. +-lft-identity 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0 99.7%

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

4. Final simplification 99.7%

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

Alternative 10: 67.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;1 - \frac{\frac{1}{x \cdot \left(x \cdot -81\right)}}{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+152}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{0.012345679012345678}{x}}{x}}{\frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e+136)
   (- 1.0 (/ (/ 1.0 (* x (* x -81.0))) (/ 0.1111111111111111 x)))
   (if (<= y 1.9e+152)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (- 1.0 (/ (/ (/ 0.012345679012345678 x) x) (/ 0.1111111111111111 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+136) {
		tmp = 1.0 - ((1.0 / (x * (x * -81.0))) / (0.1111111111111111 / x));
	} else if (y <= 1.9e+152) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - (((0.012345679012345678 / x) / x) / (0.1111111111111111 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.2d+136)) then
        tmp = 1.0d0 - ((1.0d0 / (x * (x * (-81.0d0)))) / (0.1111111111111111d0 / x))
    else if (y <= 1.9d+152) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 - (((0.012345679012345678d0 / x) / x) / (0.1111111111111111d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+136) {
		tmp = 1.0 - ((1.0 / (x * (x * -81.0))) / (0.1111111111111111 / x));
	} else if (y <= 1.9e+152) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - (((0.012345679012345678 / x) / x) / (0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e+136:
		tmp = 1.0 - ((1.0 / (x * (x * -81.0))) / (0.1111111111111111 / x))
	elif y <= 1.9e+152:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 - (((0.012345679012345678 / x) / x) / (0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e+136)
		tmp = Float64(1.0 - Float64(Float64(1.0 / Float64(x * Float64(x * -81.0))) / Float64(0.1111111111111111 / x)));
	elseif (y <= 1.9e+152)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.012345679012345678 / x) / x) / Float64(0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e+136)
		tmp = 1.0 - ((1.0 / (x * (x * -81.0))) / (0.1111111111111111 / x));
	elseif (y <= 1.9e+152)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 - (((0.012345679012345678 / x) / x) / (0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e+136], N[(1.0 - N[(N[(1.0 / N[(x * N[(x * -81.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+152], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision] / N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1999999999999999e136

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 2.8%

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

      1. div-inv 2.8%

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

      2. add0 2.8%

         \[\leadsto 1 - \color{blue}{\left(\frac{0.1111111111111111}{x} + 0\right)} \]

      3. flip-+ 2.7%

         \[\leadsto 1 - \color{blue}{\frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0}} \]

      4. div-inv 2.7%

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

      5. *-commutative 2.7%

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

      6. div-inv 2.7%

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

      7. *-commutative 2.7%

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

      8. swap-sqr 2.7%

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

      9. inv-pow 2.7%

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

      10. inv-pow 2.7%

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

      11. pow-prod-up 2.7%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-1 + -1\right)}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      12. metadata-eval 2.7%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-2}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      13. metadata-eval 2.7%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot \color{blue}{0.012345679012345678} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      14. metadata-eval 2.7%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - \color{blue}{0}}{\frac{0.1111111111111111}{x} - 0} \]

   7. Applied egg-rr 2.7%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\frac{0.1111111111111111}{x} - 0}} \]
   8. Step-by-step derivation

      1. --rgt-identity 2.7%

         \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

      2. --rgt-identity 2.7%

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

   9. Simplified 2.7%

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

   11. Applied egg-rr 22.1%

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

       1. *-commutative 22.1%

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

       2. associate-*l* 22.1%

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

       3. *-commutative 22.1%

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

       4. associate-*l* 22.1%

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

       5. metadata-eval 22.1%

          \[\leadsto 1 - \frac{\frac{1}{x \cdot \left(x \cdot \color{blue}{-81}\right)}}{\frac{0.1111111111111111}{x}} \]

   13. Simplified 22.1%

       \[\leadsto 1 - \frac{\color{blue}{\frac{1}{x \cdot \left(x \cdot -81\right)}}}{\frac{0.1111111111111111}{x}} \]

   if -2.1999999999999999e136 < y < 1.9e152

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 85.8%

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

      1. div-inv 85.8%

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

      2. add0 85.8%

         \[\leadsto 1 - \color{blue}{\left(\frac{0.1111111111111111}{x} + 0\right)} \]

      3. flip-+ 62.5%

         \[\leadsto 1 - \color{blue}{\frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0}} \]

      4. div-inv 62.5%

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

      5. *-commutative 62.5%

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

      6. div-inv 62.4%

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

      7. *-commutative 62.4%

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

      8. swap-sqr 62.4%

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

      9. inv-pow 62.4%

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

      10. inv-pow 62.4%

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

      11. pow-prod-up 62.5%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-1 + -1\right)}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      12. metadata-eval 62.5%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-2}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      13. metadata-eval 62.5%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot \color{blue}{0.012345679012345678} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      14. metadata-eval 62.5%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - \color{blue}{0}}{\frac{0.1111111111111111}{x} - 0} \]

   7. Applied egg-rr 62.5%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\frac{0.1111111111111111}{x} - 0}} \]
   8. Step-by-step derivation

      1. --rgt-identity 62.5%

         \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

      2. --rgt-identity 62.5%

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

   9. Simplified 62.5%

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

       1. add0 62.5%

          \[\leadsto 1 - \color{blue}{\left(\frac{{x}^{-2} \cdot 0.012345679012345678}{\frac{0.1111111111111111}{x}} + 0\right)} \]

       2. associate-/r/ 62.5%

          \[\leadsto 1 - \left(\color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111} \cdot x} + 0\right) \]

       3. *-commutative 62.5%

          \[\leadsto 1 - \left(\color{blue}{x \cdot \frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111}} + 0\right) \]

       4. associate-/l* 62.5%

          \[\leadsto 1 - \left(x \cdot \color{blue}{\frac{{x}^{-2}}{\frac{0.1111111111111111}{0.012345679012345678}}} + 0\right) \]

       5. metadata-eval 62.5%

          \[\leadsto 1 - \left(x \cdot \frac{{x}^{-2}}{\color{blue}{9}} + 0\right) \]

   11. Applied egg-rr 62.5%

       \[\leadsto 1 - \color{blue}{\left(x \cdot \frac{{x}^{-2}}{9} + 0\right)} \]
   12. Step-by-step derivation

       1. add0 62.5%

          \[\leadsto 1 - \color{blue}{x \cdot \frac{{x}^{-2}}{9}} \]

       2. *-commutative 62.5%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2}}{9} \cdot x} \]

       3. associate-*l/ 62.4%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot x}{9}} \]

       4. pow-plus 85.9%

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

       5. metadata-eval 85.9%

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

       6. unpow-1 85.9%

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

   13. Simplified 85.9%

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

   if 1.9e152 < 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. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 3.8%

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

      1. div-inv 3.8%

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

      2. add0 3.8%

         \[\leadsto 1 - \color{blue}{\left(\frac{0.1111111111111111}{x} + 0\right)} \]

      3. flip-+ 31.6%

         \[\leadsto 1 - \color{blue}{\frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0}} \]

      4. div-inv 31.6%

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

      5. *-commutative 31.6%

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

      6. div-inv 31.6%

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

      7. *-commutative 31.6%

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

      8. swap-sqr 31.6%

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

      9. inv-pow 31.6%

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

      10. inv-pow 31.6%

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

      11. pow-prod-up 31.6%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-1 + -1\right)}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      12. metadata-eval 31.6%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-2}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      13. metadata-eval 31.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot \color{blue}{0.012345679012345678} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      14. metadata-eval 31.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - \color{blue}{0}}{\frac{0.1111111111111111}{x} - 0} \]

   7. Applied egg-rr 31.6%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\frac{0.1111111111111111}{x} - 0}} \]
   8. Step-by-step derivation

      1. --rgt-identity 31.6%

         \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

      2. --rgt-identity 31.6%

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

   9. Simplified 31.6%

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

       1. *-commutative 31.6%

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

       2. sqr-pow 31.6%

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

       3. metadata-eval 31.6%

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

       4. swap-sqr 31.6%

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

       5. metadata-eval 31.6%

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

       6. inv-pow 31.6%

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

       7. div-inv 31.6%

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

       8. metadata-eval 31.6%

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

       9. inv-pow 31.6%

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

       10. div-inv 31.6%

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

       11. associate-*r/ 31.6%

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

   11. Applied egg-rr 31.6%

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

       1. associate-*l/ 31.6%

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

       2. metadata-eval 31.6%

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

   13. Simplified 31.6%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+136}:\\ \;\;\;\;1 - \frac{\frac{1}{x \cdot \left(x \cdot -81\right)}}{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+152}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{0.012345679012345678}{x}}{x}}{\frac{0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+152}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{0.012345679012345678}{x}}{x}}{\frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e+152)
   (+ 1.0 (/ (/ -1.0 x) 9.0))
   (- 1.0 (/ (/ (/ 0.012345679012345678 x) x) (/ 0.1111111111111111 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.9e+152) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - (((0.012345679012345678 / x) / x) / (0.1111111111111111 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.9d+152) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 - (((0.012345679012345678d0 / x) / x) / (0.1111111111111111d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e+152) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - (((0.012345679012345678 / x) / x) / (0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.9e+152:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 - (((0.012345679012345678 / x) / x) / (0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.9e+152)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.012345679012345678 / x) / x) / Float64(0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e+152)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 - (((0.012345679012345678 / x) / x) / (0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.9e+152], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision] / N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e152

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 71.7%

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

      1. div-inv 71.7%

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

      2. add0 71.7%

         \[\leadsto 1 - \color{blue}{\left(\frac{0.1111111111111111}{x} + 0\right)} \]

      3. flip-+ 52.3%

         \[\leadsto 1 - \color{blue}{\frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0}} \]

      4. div-inv 52.3%

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

      5. *-commutative 52.3%

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

      6. div-inv 52.3%

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

      7. *-commutative 52.3%

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

      8. swap-sqr 52.2%

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

      9. inv-pow 52.2%

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

      10. inv-pow 52.2%

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

      11. pow-prod-up 52.3%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-1 + -1\right)}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      12. metadata-eval 52.3%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-2}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      13. metadata-eval 52.3%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot \color{blue}{0.012345679012345678} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      14. metadata-eval 52.3%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - \color{blue}{0}}{\frac{0.1111111111111111}{x} - 0} \]

   7. Applied egg-rr 52.3%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\frac{0.1111111111111111}{x} - 0}} \]
   8. Step-by-step derivation

      1. --rgt-identity 52.3%

         \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

      2. --rgt-identity 52.3%

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

   9. Simplified 52.3%

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

       1. add0 52.3%

          \[\leadsto 1 - \color{blue}{\left(\frac{{x}^{-2} \cdot 0.012345679012345678}{\frac{0.1111111111111111}{x}} + 0\right)} \]

       2. associate-/r/ 52.3%

          \[\leadsto 1 - \left(\color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111} \cdot x} + 0\right) \]

       3. *-commutative 52.3%

          \[\leadsto 1 - \left(\color{blue}{x \cdot \frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111}} + 0\right) \]

       4. associate-/l* 52.3%

          \[\leadsto 1 - \left(x \cdot \color{blue}{\frac{{x}^{-2}}{\frac{0.1111111111111111}{0.012345679012345678}}} + 0\right) \]

       5. metadata-eval 52.3%

          \[\leadsto 1 - \left(x \cdot \frac{{x}^{-2}}{\color{blue}{9}} + 0\right) \]

   11. Applied egg-rr 52.3%

       \[\leadsto 1 - \color{blue}{\left(x \cdot \frac{{x}^{-2}}{9} + 0\right)} \]
   12. Step-by-step derivation

       1. add0 52.3%

          \[\leadsto 1 - \color{blue}{x \cdot \frac{{x}^{-2}}{9}} \]

       2. *-commutative 52.3%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2}}{9} \cdot x} \]

       3. associate-*l/ 52.3%

          \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot x}{9}} \]

       4. pow-plus 71.7%

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

       5. metadata-eval 71.7%

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

       6. unpow-1 71.7%

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

   13. Simplified 71.7%

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

   if 1.9e152 < 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. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 3.8%

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

      1. div-inv 3.8%

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

      2. add0 3.8%

         \[\leadsto 1 - \color{blue}{\left(\frac{0.1111111111111111}{x} + 0\right)} \]

      3. flip-+ 31.6%

         \[\leadsto 1 - \color{blue}{\frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0}} \]

      4. div-inv 31.6%

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

      5. *-commutative 31.6%

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

      6. div-inv 31.6%

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

      7. *-commutative 31.6%

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

      8. swap-sqr 31.6%

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

      9. inv-pow 31.6%

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

      10. inv-pow 31.6%

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

      11. pow-prod-up 31.6%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-1 + -1\right)}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      12. metadata-eval 31.6%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-2}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      13. metadata-eval 31.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot \color{blue}{0.012345679012345678} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

      14. metadata-eval 31.6%

          \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - \color{blue}{0}}{\frac{0.1111111111111111}{x} - 0} \]

   7. Applied egg-rr 31.6%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\frac{0.1111111111111111}{x} - 0}} \]
   8. Step-by-step derivation

      1. --rgt-identity 31.6%

         \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

      2. --rgt-identity 31.6%

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

   9. Simplified 31.6%

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

       1. *-commutative 31.6%

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

       2. sqr-pow 31.6%

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

       3. metadata-eval 31.6%

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

       4. swap-sqr 31.6%

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

       5. metadata-eval 31.6%

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

       6. inv-pow 31.6%

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

       7. div-inv 31.6%

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

       8. metadata-eval 31.6%

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

       9. inv-pow 31.6%

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

       10. div-inv 31.6%

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

       11. associate-*r/ 31.6%

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

   11. Applied egg-rr 31.6%

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

       1. associate-*l/ 31.6%

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

       2. metadata-eval 31.6%

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

   13. Simplified 31.6%

       \[\leadsto 1 - \frac{\color{blue}{\frac{\frac{0.012345679012345678}{x}}{x}}}{\frac{0.1111111111111111}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+152}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{\frac{0.012345679012345678}{x}}{x}}{\frac{0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.0% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = (1.0 / x) * -0.1111111111111111
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 63.0%

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

      1. clear-num 62.9%

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

      2. associate-/r/ 63.0%

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

   7. Applied egg-rr 63.0%

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

   if 0.110000000000000001 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 64.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{1}{x} \cdot -0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11) (/ -0.1111111111111111 x) 1.0))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 63.0%

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

   if 0.110000000000000001 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 64.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.0% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + 0.1111111111111111 \cdot \frac{-1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 + (0.1111111111111111 * (-1.0 / x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 64.5%

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

6. Final simplification 64.5%

   \[\leadsto 1 + 0.1111111111111111 \cdot \frac{-1}{x} \]
7. Add Preprocessing

Alternative 15: 62.1% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\frac{-1}{x}}{9} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 + ((-1.0 / x) / 9.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 64.5%

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

   1. div-inv 64.5%

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

   2. add0 64.5%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.1111111111111111}{x} + 0\right)} \]

   3. flip-+ 50.1%

      \[\leadsto 1 - \color{blue}{\frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0}} \]

   4. div-inv 50.1%

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

   5. *-commutative 50.1%

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

   6. div-inv 50.1%

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

   7. *-commutative 50.1%

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

   8. swap-sqr 50.1%

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

   9. inv-pow 50.1%

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

   10. inv-pow 50.1%

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

   11. pow-prod-up 50.1%

       \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-1 + -1\right)}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

   12. metadata-eval 50.1%

       \[\leadsto 1 - \frac{{x}^{\color{blue}{-2}} \cdot \left(0.1111111111111111 \cdot 0.1111111111111111\right) - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

   13. metadata-eval 50.1%

       \[\leadsto 1 - \frac{{x}^{-2} \cdot \color{blue}{0.012345679012345678} - 0 \cdot 0}{\frac{0.1111111111111111}{x} - 0} \]

   14. metadata-eval 50.1%

       \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - \color{blue}{0}}{\frac{0.1111111111111111}{x} - 0} \]

7. Applied egg-rr 50.1%

   \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\frac{0.1111111111111111}{x} - 0}} \]
8. Step-by-step derivation

   1. --rgt-identity 50.1%

      \[\leadsto 1 - \frac{{x}^{-2} \cdot 0.012345679012345678 - 0}{\color{blue}{\frac{0.1111111111111111}{x}}} \]

   2. --rgt-identity 50.1%

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

9. Simplified 50.1%

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

    1. add0 50.1%

       \[\leadsto 1 - \color{blue}{\left(\frac{{x}^{-2} \cdot 0.012345679012345678}{\frac{0.1111111111111111}{x}} + 0\right)} \]

    2. associate-/r/ 50.1%

       \[\leadsto 1 - \left(\color{blue}{\frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111} \cdot x} + 0\right) \]

    3. *-commutative 50.1%

       \[\leadsto 1 - \left(\color{blue}{x \cdot \frac{{x}^{-2} \cdot 0.012345679012345678}{0.1111111111111111}} + 0\right) \]

    4. associate-/l* 50.1%

       \[\leadsto 1 - \left(x \cdot \color{blue}{\frac{{x}^{-2}}{\frac{0.1111111111111111}{0.012345679012345678}}} + 0\right) \]

    5. metadata-eval 50.1%

       \[\leadsto 1 - \left(x \cdot \frac{{x}^{-2}}{\color{blue}{9}} + 0\right) \]

11. Applied egg-rr 50.1%

    \[\leadsto 1 - \color{blue}{\left(x \cdot \frac{{x}^{-2}}{9} + 0\right)} \]
12. Step-by-step derivation

    1. add0 50.1%

       \[\leadsto 1 - \color{blue}{x \cdot \frac{{x}^{-2}}{9}} \]

    2. *-commutative 50.1%

       \[\leadsto 1 - \color{blue}{\frac{{x}^{-2}}{9} \cdot x} \]

    3. associate-*l/ 50.1%

       \[\leadsto 1 - \color{blue}{\frac{{x}^{-2} \cdot x}{9}} \]

    4. pow-plus 64.5%

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

    5. metadata-eval 64.5%

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

    6. unpow-1 64.5%

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

13. Simplified 64.5%

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

14. Final simplification 64.5%

    \[\leadsto 1 + \frac{\frac{-1}{x}}{9} \]
15. Add Preprocessing

Alternative 16: 62.0% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 64.5%

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

   1. cancel-sign-sub-inv 64.5%

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

   2. metadata-eval 64.5%

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

   3. associate-*r/ 64.5%

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

   4. metadata-eval 64.5%

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

7. Simplified 64.5%

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

8. Final simplification 64.5%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]
9. Add Preprocessing

Alternative 17: 31.1% 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. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around inf 33.4%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 33.4%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024034 
(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)))))