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

Percentage Accurate: 99.7% → 99.7%

Time: 13.2s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

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]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0 99.6%

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

   1. div-inv 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. associate-*r* 99.5%

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

   4. inv-pow 99.5%

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

   5. sqrt-pow1 99.5%

      \[\leadsto \left(1 - \left(0.1111111111111111 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot \sqrt{\frac{1}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   6. metadata-eval 99.5%

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

   7. inv-pow 99.5%

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

   8. sqrt-pow1 99.5%

      \[\leadsto \left(1 - \left(0.1111111111111111 \cdot {x}^{-0.5}\right) \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   9. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

   \[\leadsto \left(1 - \color{blue}{\left(0.1111111111111111 \cdot {x}^{-0.5}\right) \cdot {x}^{-0.5}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
6. Step-by-step derivation

   1. associate-*l* 99.6%

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

   2. metadata-eval 99.6%

      \[\leadsto \left(1 - \color{blue}{{9}^{-1}} \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   3. pow-prod-up 99.6%

      \[\leadsto \left(1 - {9}^{-1} \cdot \color{blue}{{x}^{\left(-0.5 + -0.5\right)}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

   4. metadata-eval 99.6%

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

   5. unpow-prod-down 99.7%

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

   6. *-commutative 99.7%

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

   7. unpow-1 99.7%

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

   8. associate-/r* 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+40} \lor \neg \left(y \leq 5.9 \cdot 10^{+44}\right):\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.26e+40) (not (<= y 5.9e+44)))
   (* y (* -0.3333333333333333 (sqrt (/ 1.0 x))))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.26e+40) || !(y <= 5.9e+44)) {
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.26d+40)) .or. (.not. (y <= 5.9d+44))) then
        tmp = y * ((-0.3333333333333333d0) * sqrt((1.0d0 / x)))
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.26e+40) || !(y <= 5.9e+44)) {
		tmp = y * (-0.3333333333333333 * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.26e+40) or not (y <= 5.9e+44):
		tmp = y * (-0.3333333333333333 * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.26e+40) || !(y <= 5.9e+44))
		tmp = Float64(y * Float64(-0.3333333333333333 * sqrt(Float64(1.0 / 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 <= -1.26e+40) || ~((y <= 5.9e+44)))
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.26e+40], N[Not[LessEqual[y, 5.9e+44]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2599999999999999e40 or 5.89999999999999965e44 < y

   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 99.6%

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

   4. Taylor expanded in y around inf 85.6%

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

      1. associate-*r* 85.7%

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

      2. *-commutative 85.7%

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

   6. Simplified 85.7%

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

   if -1.2599999999999999e40 < y < 5.89999999999999965e44

   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 99.6%

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

      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 96.8%

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

      1. *-commutative 96.8%

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

      2. metadata-eval 96.8%

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

      3. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+40} \lor \neg \left(y \leq 5.9 \cdot 10^{+44}\right):\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+39} \lor \neg \left(y \leq 1.62 \cdot 10^{+40}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8e+39) (not (<= y 1.62e+40)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8e+39) || !(y <= 1.62e+40)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8d+39)) .or. (.not. (y <= 1.62d+40))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8e+39) || !(y <= 1.62e+40)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8e+39) or not (y <= 1.62e+40):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8e+39) || !(y <= 1.62e+40))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8e+39) || ~((y <= 1.62e+40)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8e+39], N[Not[LessEqual[y, 1.62e+40]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.99999999999999952e39 or 1.62e40 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 92.9%

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

      1. *-commutative 92.9%

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

      2. associate-*l* 93.0%

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

      3. *-commutative 93.0%

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

   7. Simplified 93.0%

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

      1. *-commutative 93.0%

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

      2. associate-*r* 92.9%

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

      3. sqrt-div 92.9%

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

      4. metadata-eval 92.9%

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

      5. div-inv 93.0%

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

      6. add-sqr-sqrt 47.8%

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

      7. unpow2 47.8%

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

      8. expm1-log1p-u 2.6%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot {\left(\sqrt{\frac{y}{\sqrt{x}}}\right)}^{2}\right)\right)} \]

      9. expm1-udef 2.6%

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

   9. Applied egg-rr 44.9%

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

       1. expm1-def 44.9%

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

       2. expm1-log1p 92.9%

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

       3. associate-*r/ 93.0%

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

       4. associate-*l/ 93.0%

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

       5. *-commutative 93.0%

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

   11. Simplified 93.0%

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

   if -7.99999999999999952e39 < y < 1.62e40

   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 99.6%

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

      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 96.8%

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

      1. *-commutative 96.8%

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

      2. metadata-eval 96.8%

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

      3. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+39} \lor \neg \left(y \leq 1.62 \cdot 10^{+40}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+39} \lor \neg \left(y \leq 4.3 \cdot 10^{+43}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.2e+39) (not (<= y 4.3e+43)))
   (+ 1.0 (/ y (* (sqrt x) -3.0)))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.2e+39) || !(y <= 4.3e+43)) {
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.2d+39)) .or. (.not. (y <= 4.3d+43))) then
        tmp = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.2e+39) || !(y <= 4.3e+43)) {
		tmp = 1.0 + (y / (Math.sqrt(x) * -3.0));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.2e+39) or not (y <= 4.3e+43):
		tmp = 1.0 + (y / (math.sqrt(x) * -3.0))
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.2e+39) || !(y <= 4.3e+43))
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.2e+39) || ~((y <= 4.3e+43)))
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.2e+39], N[Not[LessEqual[y, 4.3e+43]], $MachinePrecision]], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $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 < -7.19999999999999969e39 or 4.3e43 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 92.9%

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

      1. *-commutative 92.9%

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

      2. associate-*l* 93.0%

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

      3. *-commutative 93.0%

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

   7. Simplified 93.0%

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

      1. *-commutative 93.0%

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

      2. sqrt-div 93.0%

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

      3. metadata-eval 93.0%

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

      4. div-inv 93.0%

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

      5. *-commutative 93.0%

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

      6. associate-/l* 93.0%

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

   9. Applied egg-rr 93.0%

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

       1. add-log-exp 14.5%

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

       2. *-un-lft-identity 14.5%

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

       3. log-prod 14.5%

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

       4. metadata-eval 14.5%

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

       5. add-log-exp 93.0%

          \[\leadsto 1 + \frac{y}{0 + \color{blue}{\frac{\sqrt{x}}{-0.3333333333333333}}} \]

       6. div-inv 93.1%

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

       7. metadata-eval 93.1%

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

   11. Applied egg-rr 93.1%

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

       1. +-lft-identity 93.1%

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

   13. Simplified 93.1%

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

   if -7.19999999999999969e39 < y < 4.3e43

   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 99.6%

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

      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 96.8%

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

      1. *-commutative 96.8%

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

      2. metadata-eval 96.8%

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

      3. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+39} \lor \neg \left(y \leq 4.3 \cdot 10^{+43}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \mathbf{if}\;y \leq -1.26 \cdot 10^{+40}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot t_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (if (<= y -1.26e+40)
     (* (* y -0.3333333333333333) t_0)
     (if (<= y 3e+42)
       (+ 1.0 (/ (/ -1.0 x) 9.0))
       (* y (* -0.3333333333333333 t_0))))))

C

double code(double x, double y) {
	double t_0 = sqrt((1.0 / x));
	double tmp;
	if (y <= -1.26e+40) {
		tmp = (y * -0.3333333333333333) * t_0;
	} else if (y <= 3e+42) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = y * (-0.3333333333333333 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / x))
    if (y <= (-1.26d+40)) then
        tmp = (y * (-0.3333333333333333d0)) * t_0
    else if (y <= 3d+42) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = y * ((-0.3333333333333333d0) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((1.0 / x));
	double tmp;
	if (y <= -1.26e+40) {
		tmp = (y * -0.3333333333333333) * t_0;
	} else if (y <= 3e+42) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = y * (-0.3333333333333333 * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((1.0 / x))
	tmp = 0
	if y <= -1.26e+40:
		tmp = (y * -0.3333333333333333) * t_0
	elif y <= 3e+42:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = y * (-0.3333333333333333 * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(1.0 / x))
	tmp = 0.0
	if (y <= -1.26e+40)
		tmp = Float64(Float64(y * -0.3333333333333333) * t_0);
	elseif (y <= 3e+42)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((1.0 / x));
	tmp = 0.0;
	if (y <= -1.26e+40)
		tmp = (y * -0.3333333333333333) * t_0;
	elseif (y <= 3e+42)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = y * (-0.3333333333333333 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.26e+40], N[(N[(y * -0.3333333333333333), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 3e+42], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2599999999999999e40

   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 99.6%

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

      1. div-inv 99.6%

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

      2. add-sqr-sqrt 99.6%

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

      3. associate-*r* 99.6%

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

      4. inv-pow 99.6%

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

      5. sqrt-pow1 99.6%

         \[\leadsto \left(1 - \left(0.1111111111111111 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot \sqrt{\frac{1}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

      6. metadata-eval 99.6%

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

      7. inv-pow 99.6%

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

      8. sqrt-pow1 99.6%

         \[\leadsto \left(1 - \left(0.1111111111111111 \cdot {x}^{-0.5}\right) \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

      9. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

      \[\leadsto \left(1 - \color{blue}{\left(0.1111111111111111 \cdot {x}^{-0.5}\right) \cdot {x}^{-0.5}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. associate-*l* 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \left(1 - \color{blue}{{9}^{-1}} \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right) - \frac{y}{3 \cdot \sqrt{x}} \]

      3. pow-prod-up 99.6%

         \[\leadsto \left(1 - {9}^{-1} \cdot \color{blue}{{x}^{\left(-0.5 + -0.5\right)}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

      4. metadata-eval 99.6%

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

      5. unpow-prod-down 99.6%

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

      6. *-commutative 99.6%

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

      7. unpow-1 99.6%

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

      8. associate-/r* 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in y around inf 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 84.9%

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

      3. *-commutative 84.9%

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

   10. Simplified 84.9%

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

   if -1.2599999999999999e40 < y < 3.00000000000000029e42

   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 99.6%

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

      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 96.8%

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

      1. *-commutative 96.8%

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

      2. metadata-eval 96.8%

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

      3. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

   if 3.00000000000000029e42 < y

   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 99.5%

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

   4. Taylor expanded in y around inf 86.4%

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

      1. associate-*r* 86.5%

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

      2. *-commutative 86.5%

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

   6. Simplified 86.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+40}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+39}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.3e+39)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 4.1e+40)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.3e+39) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 4.1e+40) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.3d+39)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 4.1d+40) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.3e+39) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 4.1e+40) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.3e+39:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 4.1e+40:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.3e+39)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 4.1e+40)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.3e+39)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 4.1e+40)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.3e+39], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+40], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.29999999999999979e39

   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. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*l* 92.5%

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

      3. *-commutative 92.5%

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

   7. Simplified 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-*r* 92.4%

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

      3. sqrt-div 92.4%

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

      4. metadata-eval 92.4%

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

      5. div-inv 92.5%

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

      6. add-sqr-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. expm1-log1p-u 0.0%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot {\left(\sqrt{\frac{y}{\sqrt{x}}}\right)}^{2}\right)\right)} \]

      9. expm1-udef 0.0%

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

   9. Applied egg-rr 86.8%

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

       1. expm1-def 86.8%

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

       2. expm1-log1p 92.4%

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

       3. associate-*r/ 92.4%

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

       4. associate-*l/ 92.5%

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

       5. *-commutative 92.5%

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

   11. Simplified 92.5%

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

   if -5.29999999999999979e39 < y < 4.1000000000000002e40

   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 99.6%

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

      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 96.8%

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

      1. *-commutative 96.8%

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

      2. metadata-eval 96.8%

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

      3. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

   if 4.1000000000000002e40 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. associate-*l* 93.5%

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

      3. *-commutative 93.5%

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

   7. Simplified 93.5%

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

      1. *-commutative 93.5%

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

      2. associate-*r* 93.3%

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

      3. sqrt-div 93.4%

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

      4. metadata-eval 93.4%

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

      5. div-inv 93.4%

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

      6. add-sqr-sqrt 93.2%

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

      7. unpow2 93.2%

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

      8. expm1-log1p-u 5.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot {\left(\sqrt{\frac{y}{\sqrt{x}}}\right)}^{2}\right)\right)} \]

      9. expm1-udef 5.1%

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

   9. Applied egg-rr 5.1%

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

       1. expm1-def 5.1%

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

       2. expm1-log1p 93.4%

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

       3. *-commutative 93.4%

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

       4. associate-/r/ 93.5%

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

   11. Simplified 93.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+39}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 4.25 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.06e+40)
   (+ 1.0 (/ y (* (sqrt x) -3.0)))
   (if (<= y 4.25e+40)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (+ 1.0 (/ (/ y -3.0) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+40) {
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	} else if (y <= 4.25e+40) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.06d+40)) then
        tmp = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    else if (y <= 4.25d+40) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+40) {
		tmp = 1.0 + (y / (Math.sqrt(x) * -3.0));
	} else if (y <= 4.25e+40) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.06e+40:
		tmp = 1.0 + (y / (math.sqrt(x) * -3.0))
	elif y <= 4.25e+40:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.06e+40)
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	elseif (y <= 4.25e+40)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.06e+40)
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	elseif (y <= 4.25e+40)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.06e+40], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.25e+40], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05999999999999996e40

   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. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*l* 92.5%

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

      3. *-commutative 92.5%

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

   7. Simplified 92.5%

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

      1. *-commutative 92.5%

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

      2. sqrt-div 92.5%

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

      3. metadata-eval 92.5%

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

      4. div-inv 92.4%

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

      5. *-commutative 92.4%

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

      6. associate-/l* 92.5%

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

   9. Applied egg-rr 92.5%

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

       1. add-log-exp 14.4%

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

       2. *-un-lft-identity 14.4%

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

       3. log-prod 14.4%

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

       4. metadata-eval 14.4%

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

       5. add-log-exp 92.5%

          \[\leadsto 1 + \frac{y}{0 + \color{blue}{\frac{\sqrt{x}}{-0.3333333333333333}}} \]

       6. div-inv 92.6%

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

       7. metadata-eval 92.6%

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

   11. Applied egg-rr 92.6%

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

       1. +-lft-identity 92.6%

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

   13. Simplified 92.6%

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

   if -1.05999999999999996e40 < y < 4.24999999999999998e40

   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 99.6%

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

      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 96.8%

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

      1. *-commutative 96.8%

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

      2. metadata-eval 96.8%

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

      3. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

   if 4.24999999999999998e40 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. associate-*l* 93.5%

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

      3. *-commutative 93.5%

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

   7. Simplified 93.5%

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

      1. add-cbrt-cube 86.0%

         \[\leadsto 1 + \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right) \cdot \sqrt{\frac{1}{x}}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      2. pow1/3 82.3%

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

      3. add-sqr-sqrt 82.3%

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

      4. pow1 82.3%

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

      5. pow1/2 82.3%

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

      6. pow-prod-up 82.3%

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

      7. metadata-eval 82.3%

         \[\leadsto 1 + {\left({\left(\frac{1}{x}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   9. Applied egg-rr 82.3%

      \[\leadsto 1 + \color{blue}{{\left({\left(\frac{1}{x}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   10. Step-by-step derivation

       1. unpow1/3 86.1%

          \[\leadsto 1 + \color{blue}{\sqrt[3]{{\left(\frac{1}{x}\right)}^{1.5}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   11. Simplified 86.1%

       \[\leadsto 1 + \color{blue}{\sqrt[3]{{\left(\frac{1}{x}\right)}^{1.5}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   12. Step-by-step derivation

       1. pow1/3 82.3%

          \[\leadsto 1 + \color{blue}{{\left({\left(\frac{1}{x}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       2. pow-pow 93.5%

          \[\leadsto 1 + \color{blue}{{\left(\frac{1}{x}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       3. metadata-eval 93.5%

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

       4. pow1/2 93.5%

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

       5. inv-pow 93.5%

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

       6. sqrt-pow1 93.6%

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

       7. metadata-eval 93.6%

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

       8. sqr-pow 93.2%

          \[\leadsto 1 + \color{blue}{\left({x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       9. pow2 93.2%

          \[\leadsto 1 + \color{blue}{{\left({x}^{\left(\frac{-0.5}{2}\right)}\right)}^{2}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       10. metadata-eval 93.2%

           \[\leadsto 1 + {\left({x}^{\color{blue}{-0.25}}\right)}^{2} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   13. Applied egg-rr 93.2%

       \[\leadsto 1 + \color{blue}{{\left({x}^{-0.25}\right)}^{2}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   14. Step-by-step derivation

       1. *-commutative 93.2%

          \[\leadsto 1 + \color{blue}{\left(-0.3333333333333333 \cdot y\right) \cdot {\left({x}^{-0.25}\right)}^{2}} \]

       2. associate-*l* 93.3%

          \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(y \cdot {\left({x}^{-0.25}\right)}^{2}\right)} \]

       3. metadata-eval 93.3%

          \[\leadsto 1 + \color{blue}{\frac{1}{-3}} \cdot \left(y \cdot {\left({x}^{-0.25}\right)}^{2}\right) \]

       4. pow-pow 93.4%

          \[\leadsto 1 + \frac{1}{-3} \cdot \left(y \cdot \color{blue}{{x}^{\left(-0.25 \cdot 2\right)}}\right) \]

       5. metadata-eval 93.4%

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

       6. metadata-eval 93.4%

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

       7. sqrt-pow1 93.3%

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

       8. inv-pow 93.3%

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

       9. sqrt-div 93.4%

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

       10. metadata-eval 93.4%

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

       11. div-inv 93.4%

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

       12. times-frac 93.6%

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

       13. *-un-lft-identity 93.6%

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

       14. associate-/r* 93.6%

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

   15. Applied egg-rr 93.6%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 4.25 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.5e+39)
   (+ 1.0 (/ (/ y (sqrt x)) -3.0))
   (if (<= y 9.2e+40)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (+ 1.0 (/ (/ y -3.0) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+39) {
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	} else if (y <= 9.2e+40) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.5d+39)) then
        tmp = 1.0d0 + ((y / sqrt(x)) / (-3.0d0))
    else if (y <= 9.2d+40) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+39) {
		tmp = 1.0 + ((y / Math.sqrt(x)) / -3.0);
	} else if (y <= 9.2e+40) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.5e+39:
		tmp = 1.0 + ((y / math.sqrt(x)) / -3.0)
	elif y <= 9.2e+40:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5e+39)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) / -3.0));
	elseif (y <= 9.2e+40)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.5e+39)
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	elseif (y <= 9.2e+40)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5e+39], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e+40], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.49999999999999996e39

   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. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*l* 92.5%

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

      3. *-commutative 92.5%

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

   7. Simplified 92.5%

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

      1. add-cbrt-cube 83.8%

         \[\leadsto 1 + \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right) \cdot \sqrt{\frac{1}{x}}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      2. pow1/3 79.9%

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

      3. add-sqr-sqrt 79.9%

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

      4. pow1 79.9%

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

      5. pow1/2 79.9%

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

      6. pow-prod-up 79.8%

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

      7. metadata-eval 79.8%

         \[\leadsto 1 + {\left({\left(\frac{1}{x}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   9. Applied egg-rr 79.8%

      \[\leadsto 1 + \color{blue}{{\left({\left(\frac{1}{x}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   10. Step-by-step derivation

       1. unpow1/3 83.8%

          \[\leadsto 1 + \color{blue}{\sqrt[3]{{\left(\frac{1}{x}\right)}^{1.5}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   11. Simplified 83.8%

       \[\leadsto 1 + \color{blue}{\sqrt[3]{{\left(\frac{1}{x}\right)}^{1.5}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   12. Step-by-step derivation

       1. associate-*r* 83.8%

          \[\leadsto 1 + \color{blue}{\left(\sqrt[3]{{\left(\frac{1}{x}\right)}^{1.5}} \cdot -0.3333333333333333\right) \cdot y} \]

       2. pow1/3 79.8%

          \[\leadsto 1 + \left(\color{blue}{{\left({\left(\frac{1}{x}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot -0.3333333333333333\right) \cdot y \]

       3. pow-pow 92.5%

          \[\leadsto 1 + \left(\color{blue}{{\left(\frac{1}{x}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot -0.3333333333333333\right) \cdot y \]

       4. metadata-eval 92.5%

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

       5. pow1/2 92.5%

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

       6. sqrt-div 92.4%

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

       7. metadata-eval 92.4%

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

       8. associate-/r/ 92.4%

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

       9. associate-/r/ 92.5%

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

       10. clear-num 92.5%

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

       11. div-inv 92.6%

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

       12. associate-/r* 92.6%

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

       13. metadata-eval 92.6%

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

   13. Applied egg-rr 92.6%

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

   if -4.49999999999999996e39 < y < 9.19999999999999975e40

   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 99.6%

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

      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 96.8%

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

      1. *-commutative 96.8%

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

      2. metadata-eval 96.8%

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

      3. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

   if 9.19999999999999975e40 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. associate-*l* 93.5%

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

      3. *-commutative 93.5%

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

   7. Simplified 93.5%

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

      1. add-cbrt-cube 86.0%

         \[\leadsto 1 + \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right) \cdot \sqrt{\frac{1}{x}}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      2. pow1/3 82.3%

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

      3. add-sqr-sqrt 82.3%

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

      4. pow1 82.3%

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

      5. pow1/2 82.3%

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

      6. pow-prod-up 82.3%

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

      7. metadata-eval 82.3%

         \[\leadsto 1 + {\left({\left(\frac{1}{x}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   9. Applied egg-rr 82.3%

      \[\leadsto 1 + \color{blue}{{\left({\left(\frac{1}{x}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   10. Step-by-step derivation

       1. unpow1/3 86.1%

          \[\leadsto 1 + \color{blue}{\sqrt[3]{{\left(\frac{1}{x}\right)}^{1.5}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   11. Simplified 86.1%

       \[\leadsto 1 + \color{blue}{\sqrt[3]{{\left(\frac{1}{x}\right)}^{1.5}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   12. Step-by-step derivation

       1. pow1/3 82.3%

          \[\leadsto 1 + \color{blue}{{\left({\left(\frac{1}{x}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       2. pow-pow 93.5%

          \[\leadsto 1 + \color{blue}{{\left(\frac{1}{x}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       3. metadata-eval 93.5%

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

       4. pow1/2 93.5%

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

       5. inv-pow 93.5%

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

       6. sqrt-pow1 93.6%

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

       7. metadata-eval 93.6%

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

       8. sqr-pow 93.2%

          \[\leadsto 1 + \color{blue}{\left({x}^{\left(\frac{-0.5}{2}\right)} \cdot {x}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       9. pow2 93.2%

          \[\leadsto 1 + \color{blue}{{\left({x}^{\left(\frac{-0.5}{2}\right)}\right)}^{2}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

       10. metadata-eval 93.2%

           \[\leadsto 1 + {\left({x}^{\color{blue}{-0.25}}\right)}^{2} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   13. Applied egg-rr 93.2%

       \[\leadsto 1 + \color{blue}{{\left({x}^{-0.25}\right)}^{2}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   14. Step-by-step derivation

       1. *-commutative 93.2%

          \[\leadsto 1 + \color{blue}{\left(-0.3333333333333333 \cdot y\right) \cdot {\left({x}^{-0.25}\right)}^{2}} \]

       2. associate-*l* 93.3%

          \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(y \cdot {\left({x}^{-0.25}\right)}^{2}\right)} \]

       3. metadata-eval 93.3%

          \[\leadsto 1 + \color{blue}{\frac{1}{-3}} \cdot \left(y \cdot {\left({x}^{-0.25}\right)}^{2}\right) \]

       4. pow-pow 93.4%

          \[\leadsto 1 + \frac{1}{-3} \cdot \left(y \cdot \color{blue}{{x}^{\left(-0.25 \cdot 2\right)}}\right) \]

       5. metadata-eval 93.4%

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

       6. metadata-eval 93.4%

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

       7. sqrt-pow1 93.3%

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

       8. inv-pow 93.3%

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

       9. sqrt-div 93.4%

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

       10. metadata-eval 93.4%

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

       11. div-inv 93.4%

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

       12. times-frac 93.6%

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

       13. *-un-lft-identity 93.6%

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

       14. associate-/r* 93.6%

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

   15. Applied egg-rr 93.6%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.3e+39)
   (+ 1.0 (/ (/ y (sqrt x)) -3.0))
   (if (<= y 6.2e+42)
     (+ 1.0 (/ (/ -1.0 x) 9.0))
     (- 1.0 (/ y (sqrt (* x 9.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.3e+39) {
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	} else if (y <= 6.2e+42) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.3d+39)) then
        tmp = 1.0d0 + ((y / sqrt(x)) / (-3.0d0))
    else if (y <= 6.2d+42) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.3e+39) {
		tmp = 1.0 + ((y / Math.sqrt(x)) / -3.0);
	} else if (y <= 6.2e+42) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.3e+39:
		tmp = 1.0 + ((y / math.sqrt(x)) / -3.0)
	elif y <= 6.2e+42:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.3e+39)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) / -3.0));
	elseif (y <= 6.2e+42)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.3e+39)
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	elseif (y <= 6.2e+42)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.3e+39], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+42], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.29999999999999979e39

   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. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*l* 92.5%

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

      3. *-commutative 92.5%

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

   7. Simplified 92.5%

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

      1. add-cbrt-cube 83.8%

         \[\leadsto 1 + \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right) \cdot \sqrt{\frac{1}{x}}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      2. pow1/3 79.9%

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

      3. add-sqr-sqrt 79.9%

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

      4. pow1 79.9%

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

      5. pow1/2 79.9%

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

      6. pow-prod-up 79.8%

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

      7. metadata-eval 79.8%

         \[\leadsto 1 + {\left({\left(\frac{1}{x}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   9. Applied egg-rr 79.8%

      \[\leadsto 1 + \color{blue}{{\left({\left(\frac{1}{x}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   10. Step-by-step derivation

       1. unpow1/3 83.8%

          \[\leadsto 1 + \color{blue}{\sqrt[3]{{\left(\frac{1}{x}\right)}^{1.5}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

   11. Simplified 83.8%

       \[\leadsto 1 + \color{blue}{\sqrt[3]{{\left(\frac{1}{x}\right)}^{1.5}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
   12. Step-by-step derivation

       1. associate-*r* 83.8%

          \[\leadsto 1 + \color{blue}{\left(\sqrt[3]{{\left(\frac{1}{x}\right)}^{1.5}} \cdot -0.3333333333333333\right) \cdot y} \]

       2. pow1/3 79.8%

          \[\leadsto 1 + \left(\color{blue}{{\left({\left(\frac{1}{x}\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot -0.3333333333333333\right) \cdot y \]

       3. pow-pow 92.5%

          \[\leadsto 1 + \left(\color{blue}{{\left(\frac{1}{x}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot -0.3333333333333333\right) \cdot y \]

       4. metadata-eval 92.5%

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

       5. pow1/2 92.5%

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

       6. sqrt-div 92.4%

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

       7. metadata-eval 92.4%

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

       8. associate-/r/ 92.4%

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

       9. associate-/r/ 92.5%

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

       10. clear-num 92.5%

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

       11. div-inv 92.6%

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

       12. associate-/r* 92.6%

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

       13. metadata-eval 92.6%

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

   13. Applied egg-rr 92.6%

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

   if -5.29999999999999979e39 < y < 6.2000000000000003e42

   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 99.6%

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

      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 96.8%

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

      1. *-commutative 96.8%

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

      2. metadata-eval 96.8%

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

      3. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

   if 6.2000000000000003e42 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 93.3%

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

      1. *-commutative 93.3%

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

      2. associate-*l* 93.5%

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

      3. *-commutative 93.5%

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

   7. Simplified 93.5%

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

      1. *-commutative 93.5%

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

      2. sqrt-div 93.5%

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

      3. metadata-eval 93.5%

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

      4. div-inv 93.5%

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

      5. *-commutative 93.5%

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

      6. associate-/l* 93.4%

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

   9. Applied egg-rr 93.4%

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

       1. add-log-exp 14.5%

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

       2. *-un-lft-identity 14.5%

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

       3. log-prod 14.5%

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

       4. metadata-eval 14.5%

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

       5. add-log-exp 93.4%

          \[\leadsto 1 + \frac{y}{0 + \color{blue}{\frac{\sqrt{x}}{-0.3333333333333333}}} \]

       6. div-inv 93.6%

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

       7. metadata-eval 93.6%

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

   11. Applied egg-rr 93.6%

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

       1. +-lft-identity 93.6%

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

   13. Simplified 93.6%

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

       1. frac-2neg 93.6%

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

       2. *-commutative 93.6%

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

       3. distribute-lft-neg-in 93.6%

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

       4. metadata-eval 93.6%

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

       5. distribute-frac-neg 93.6%

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

       6. neg-sub0 93.6%

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

       7. add-sqr-sqrt 93.1%

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

       8. sqrt-unprod 93.6%

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

       9. *-commutative 93.6%

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

       10. *-commutative 93.6%

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

       11. swap-sqr 93.5%

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

       12. metadata-eval 93.5%

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

       13. metadata-eval 93.5%

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

       14. swap-sqr 93.6%

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

       15. swap-sqr 93.5%

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

       16. add-sqr-sqrt 93.6%

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

       17. metadata-eval 93.6%

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

   15. Applied egg-rr 93.6%

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

       1. neg-sub0 93.6%

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

       2. distribute-frac-neg 93.6%

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

   17. Simplified 93.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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]

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 11: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-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* 99.6%

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

   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 99.6%

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

   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 12: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. metadata-eval 99.7%

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

   3. div-inv 99.6%

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

   4. clear-num 99.6%

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

   5. *-un-lft-identity 99.6%

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

   6. times-frac 99.6%

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

   7. metadata-eval 99.6%

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

   8. metadata-eval 99.6%

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

   9. cancel-sign-sub-inv 99.6%

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

   10. associate-+l- 99.6%

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

   11. +-commutative 99.6%

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

   12. div-inv 99.5%

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

   13. cancel-sign-sub-inv 99.5%

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

   14. metadata-eval 99.5%

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

   15. div-inv 99.6%

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

   16. associate-*r/ 99.6%

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

   17. +-commutative 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 13: 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 99.7%

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

3. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 14: 62.3% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.9) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.9) {
		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 <= 1.9d0) 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 <= 1.9) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[x, 1.9], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   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 99.5%

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

   4. Taylor expanded in x around 0 58.5%

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

   if 1.8999999999999999 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. *-commutative 99.8%

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

      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 99.7%

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

      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 58.4%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.3% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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* 99.6%

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

   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 99.6%

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

   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 59.3%

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

   1. un-div-inv 59.3%

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

   2. metadata-eval 59.3%

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

   3. associate-/r* 59.3%

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

   4. *-commutative 59.3%

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

7. Applied egg-rr 59.3%

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

8. Final simplification 59.3%

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

Alternative 16: 63.3% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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* 99.6%

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

   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 99.6%

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

   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 59.3%

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

   1. *-commutative 59.3%

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

   2. metadata-eval 59.3%

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

   3. div-inv 59.4%

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

7. Applied egg-rr 59.4%

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

8. Final simplification 59.4%

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

Alternative 17: 63.3% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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* 99.6%

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

   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 99.6%

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

   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 59.3%

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

   1. cancel-sign-sub-inv 59.3%

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

   2. metadata-eval 59.3%

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

   3. associate-*r/ 59.3%

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

   4. metadata-eval 59.3%

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

   5. +-commutative 59.3%

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

7. Simplified 59.3%

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

8. Final simplification 59.3%

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

Alternative 18: 32.6% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. 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* 99.6%

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

   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 99.6%

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

   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 31.7%

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

6. Final simplification 31.7%

   \[\leadsto 1 \]
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 2024020 
(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)))))