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

Percentage Accurate: 99.7% → 99.7%

Time: 9.3s

Alternatives: 22

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

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 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+60} \lor \neg \left(y \leq 1.9 \cdot 10^{+42}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.5e+60) (not (<= y 1.9e+42)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e+60) || !(y <= 1.9e+42)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.5d+60)) .or. (.not. (y <= 1.9d+42))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e+60) || !(y <= 1.9e+42)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.5e+60) or not (y <= 1.9e+42):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.5e+60) || !(y <= 1.9e+42))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.5e+60) || ~((y <= 1.9e+42)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.5e+60], N[Not[LessEqual[y, 1.9e+42]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.49999999999999931e60 or 1.8999999999999999e42 < y

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-/r* 99.4%

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

      4. metadata-eval 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. neg-mul-1 99.4%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 95.6%

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

   if -6.49999999999999931e60 < y < 1.8999999999999999e42

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. metadata-eval 94.9%

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

   7. Simplified 94.9%

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

      1. metadata-eval 94.9%

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

      2. add-sqr-sqrt 94.8%

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

      3. frac-times 94.7%

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

      4. div-inv 94.7%

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

      5. div-inv 94.6%

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

      6. swap-sqr 94.7%

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

      7. metadata-eval 94.7%

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

      8. pow1/2 94.7%

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

      9. pow-flip 94.8%

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

      10. metadata-eval 94.8%

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

      11. pow1/2 94.8%

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

      12. pow-flip 94.8%

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

      13. metadata-eval 94.8%

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

   9. Applied egg-rr 94.8%

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

       1. *-commutative 94.8%

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

       2. pow-prod-up 94.9%

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

       3. metadata-eval 94.9%

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

       4. inv-pow 94.9%

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

       5. associate-/r/ 94.9%

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

       6. div-inv 95.0%

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

       7. metadata-eval 95.0%

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

   11. Applied egg-rr 95.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+60} \lor \neg \left(y \leq 1.9 \cdot 10^{+42}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+60}:\\ \;\;\;\;1 + \frac{-1}{\sqrt{x} \cdot \frac{3}{y}}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+60)
   (+ 1.0 (/ -1.0 (* (sqrt x) (/ 3.0 y))))
   (if (<= y 2.35e+42)
     (+ 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 <= -6.5e+60) {
		tmp = 1.0 + (-1.0 / (sqrt(x) * (3.0 / y)));
	} else if (y <= 2.35e+42) {
		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 <= (-6.5d+60)) then
        tmp = 1.0d0 + ((-1.0d0) / (sqrt(x) * (3.0d0 / y)))
    else if (y <= 2.35d+42) 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 <= -6.5e+60) {
		tmp = 1.0 + (-1.0 / (Math.sqrt(x) * (3.0 / y)));
	} else if (y <= 2.35e+42) {
		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 <= -6.5e+60:
		tmp = 1.0 + (-1.0 / (math.sqrt(x) * (3.0 / y)))
	elif y <= 2.35e+42:
		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 <= -6.5e+60)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(sqrt(x) * Float64(3.0 / y))));
	elseif (y <= 2.35e+42)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.5e+60)
		tmp = 1.0 + (-1.0 / (sqrt(x) * (3.0 / y)));
	elseif (y <= 2.35e+42)
		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, -6.5e+60], N[(1.0 + N[(-1.0 / N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.35e+42], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.49999999999999931e60

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-/r* 99.2%

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

      4. metadata-eval 99.2%

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

      5. distribute-frac-neg 99.2%

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

      6. neg-mul-1 99.2%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l/ 99.4%

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

      3. associate-*r/ 99.4%

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

      4. frac-2neg 99.4%

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

      5. associate-*r/ 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. div-inv 99.4%

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

      9. distribute-neg-frac2 99.4%

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

      10. associate-/r* 99.2%

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

      11. clear-num 99.3%

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

      12. distribute-neg-frac 99.3%

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

      13. metadata-eval 99.3%

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

      14. *-commutative 99.3%

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

      15. associate-/l* 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around inf 96.6%

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

   if -6.49999999999999931e60 < y < 2.34999999999999993e42

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. metadata-eval 94.9%

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

   7. Simplified 94.9%

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

      1. metadata-eval 94.9%

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

      2. add-sqr-sqrt 94.8%

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

      3. frac-times 94.7%

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

      4. div-inv 94.7%

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

      5. div-inv 94.6%

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

      6. swap-sqr 94.7%

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

      7. metadata-eval 94.7%

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

      8. pow1/2 94.7%

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

      9. pow-flip 94.8%

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

      10. metadata-eval 94.8%

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

      11. pow1/2 94.8%

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

      12. pow-flip 94.8%

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

      13. metadata-eval 94.8%

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

   9. Applied egg-rr 94.8%

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

       1. *-commutative 94.8%

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

       2. pow-prod-up 94.9%

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

       3. metadata-eval 94.9%

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

       4. inv-pow 94.9%

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

       5. associate-/r/ 94.9%

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

       6. div-inv 95.0%

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

       7. metadata-eval 95.0%

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

   11. Applied egg-rr 95.0%

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

   if 2.34999999999999993e42 < 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 x around inf 95.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+60}:\\ \;\;\;\;1 + \frac{-1}{\sqrt{x} \cdot \frac{3}{y}}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+61}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e+61)
   (+ 1.0 (* y (* -0.3333333333333333 (sqrt (/ 1.0 x)))))
   (if (<= y 1.9e+42)
     (+ 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 <= -2.7e+61) {
		tmp = 1.0 + (y * (-0.3333333333333333 * sqrt((1.0 / x))));
	} else if (y <= 1.9e+42) {
		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 <= (-2.7d+61)) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) * sqrt((1.0d0 / x))))
    else if (y <= 1.9d+42) 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 <= -2.7e+61) {
		tmp = 1.0 + (y * (-0.3333333333333333 * Math.sqrt((1.0 / x))));
	} else if (y <= 1.9e+42) {
		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 <= -2.7e+61:
		tmp = 1.0 + (y * (-0.3333333333333333 * math.sqrt((1.0 / x))))
	elif y <= 1.9e+42:
		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 <= -2.7e+61)
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 * sqrt(Float64(1.0 / x)))));
	elseif (y <= 1.9e+42)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.7e+61)
		tmp = 1.0 + (y * (-0.3333333333333333 * sqrt((1.0 / x))));
	elseif (y <= 1.9e+42)
		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, -2.7e+61], N[(1.0 + N[(y * N[(-0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+42], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7000000000000002e61

   1. Initial program 99.2%

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

      1. associate--l- 99.2%

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

      2. sub-neg 99.2%

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

      3. +-commutative 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. distribute-frac-neg 99.2%

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

      6. sub-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-/l* 99.2%

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

      10. fmm-def 99.2%

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

      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 x around inf 96.5%

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

      1. associate-*r* 96.5%

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

   7. Simplified 96.5%

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

   if -2.7000000000000002e61 < y < 1.8999999999999999e42

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. metadata-eval 94.9%

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

   7. Simplified 94.9%

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

      1. metadata-eval 94.9%

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

      2. add-sqr-sqrt 94.8%

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

      3. frac-times 94.7%

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

      4. div-inv 94.7%

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

      5. div-inv 94.6%

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

      6. swap-sqr 94.7%

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

      7. metadata-eval 94.7%

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

      8. pow1/2 94.7%

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

      9. pow-flip 94.8%

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

      10. metadata-eval 94.8%

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

      11. pow1/2 94.8%

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

      12. pow-flip 94.8%

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

      13. metadata-eval 94.8%

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

   9. Applied egg-rr 94.8%

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

       1. *-commutative 94.8%

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

       2. pow-prod-up 94.9%

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

       3. metadata-eval 94.9%

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

       4. inv-pow 94.9%

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

       5. associate-/r/ 94.9%

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

       6. div-inv 95.0%

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

       7. metadata-eval 95.0%

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

   11. Applied egg-rr 95.0%

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

   if 1.8999999999999999e42 < 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 x around inf 95.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+61}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+61}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+61)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 5.2e+42)
     (+ 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 <= -1.45e+61) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 5.2e+42) {
		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 <= (-1.45d+61)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 5.2d+42) 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 <= -1.45e+61) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 5.2e+42) {
		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 <= -1.45e+61:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 5.2e+42:
		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 <= -1.45e+61)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 5.2e+42)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+61)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 5.2e+42)
		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, -1.45e+61], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e+42], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45e61

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-/r* 99.2%

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

      4. metadata-eval 99.2%

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

      5. distribute-frac-neg 99.2%

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

      6. neg-mul-1 99.2%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 96.5%

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

   if -1.45e61 < y < 5.1999999999999998e42

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. metadata-eval 94.9%

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

   7. Simplified 94.9%

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

      1. metadata-eval 94.9%

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

      2. add-sqr-sqrt 94.8%

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

      3. frac-times 94.7%

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

      4. div-inv 94.7%

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

      5. div-inv 94.6%

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

      6. swap-sqr 94.7%

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

      7. metadata-eval 94.7%

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

      8. pow1/2 94.7%

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

      9. pow-flip 94.8%

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

      10. metadata-eval 94.8%

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

      11. pow1/2 94.8%

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

      12. pow-flip 94.8%

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

      13. metadata-eval 94.8%

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

   9. Applied egg-rr 94.8%

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

       1. *-commutative 94.8%

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

       2. pow-prod-up 94.9%

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

       3. metadata-eval 94.9%

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

       4. inv-pow 94.9%

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

       5. associate-/r/ 94.9%

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

       6. div-inv 95.0%

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

       7. metadata-eval 95.0%

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

   11. Applied egg-rr 95.0%

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

   if 5.1999999999999998e42 < 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 x around inf 95.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+61}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+60}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+41}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.5e+60)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 1.05e+41)
     (+ 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 <= -7.5e+60) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 1.05e+41) {
		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 <= (-7.5d+60)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 1.05d+41) 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 <= -7.5e+60) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 1.05e+41) {
		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 <= -7.5e+60:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 1.05e+41:
		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 <= -7.5e+60)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 1.05e+41)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(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 <= -7.5e+60)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 1.05e+41)
		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, -7.5e+60], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+41], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $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 < -7.5e60

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-/r* 99.2%

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

      4. metadata-eval 99.2%

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

      5. distribute-frac-neg 99.2%

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

      6. neg-mul-1 99.2%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 96.5%

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

   if -7.5e60 < y < 1.05e41

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. metadata-eval 94.9%

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

   7. Simplified 94.9%

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

      1. metadata-eval 94.9%

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

      2. add-sqr-sqrt 94.8%

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

      3. frac-times 94.7%

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

      4. div-inv 94.7%

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

      5. div-inv 94.6%

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

      6. swap-sqr 94.7%

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

      7. metadata-eval 94.7%

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

      8. pow1/2 94.7%

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

      9. pow-flip 94.8%

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

      10. metadata-eval 94.8%

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

      11. pow1/2 94.8%

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

      12. pow-flip 94.8%

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

      13. metadata-eval 94.8%

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

   9. Applied egg-rr 94.8%

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

       1. *-commutative 94.8%

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

       2. pow-prod-up 94.9%

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

       3. metadata-eval 94.9%

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

       4. inv-pow 94.9%

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

       5. associate-/r/ 94.9%

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

       6. div-inv 95.0%

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

       7. metadata-eval 95.0%

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

   11. Applied egg-rr 95.0%

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

   if 1.05e41 < 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. *-commutative 99.6%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 95.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+60}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+41}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+68} \lor \neg \left(y \leq 1.6 \cdot 10^{+118}\right):\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.5e+68) (not (<= y 1.6e+118)))
   (* (pow x -0.5) (* y -0.3333333333333333))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.5e+68) || !(y <= 1.6e+118)) {
		tmp = pow(x, -0.5) * (y * -0.3333333333333333);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.5d+68)) .or. (.not. (y <= 1.6d+118))) then
        tmp = (x ** (-0.5d0)) * (y * (-0.3333333333333333d0))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.5e+68) || !(y <= 1.6e+118)) {
		tmp = Math.pow(x, -0.5) * (y * -0.3333333333333333);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.5e+68) or not (y <= 1.6e+118):
		tmp = math.pow(x, -0.5) * (y * -0.3333333333333333)
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.5e+68) || !(y <= 1.6e+118))
		tmp = Float64((x ^ -0.5) * Float64(y * -0.3333333333333333));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.5e+68) || ~((y <= 1.6e+118)))
		tmp = (x ^ -0.5) * (y * -0.3333333333333333);
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.5e+68], N[Not[LessEqual[y, 1.6e+118]], $MachinePrecision]], N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5000000000000001e68 or 1.60000000000000008e118 < y

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-/r* 99.4%

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

      4. metadata-eval 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. neg-mul-1 99.4%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.3%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in x around inf 98.0%

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

   8. Taylor expanded in x around 0 95.1%

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

      1. *-commutative 95.1%

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

      2. associate-*l* 95.1%

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

      3. unpow-1 95.1%

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

      4. metadata-eval 95.1%

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

      5. pow-sqr 95.1%

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

      6. rem-sqrt-square 95.1%

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

      7. metadata-eval 95.1%

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

      8. pow-sqr 94.9%

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

      9. fabs-sqr 94.9%

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

      10. pow-sqr 95.1%

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

      11. metadata-eval 95.1%

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

      12. *-commutative 95.1%

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

   10. Simplified 95.1%

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

   if -1.5000000000000001e68 < y < 1.60000000000000008e118

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 92.2%

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

      1. associate-*r/ 92.2%

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

      2. metadata-eval 92.2%

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

   7. Simplified 92.2%

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

      1. metadata-eval 92.2%

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

      2. add-sqr-sqrt 92.1%

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

      3. frac-times 92.0%

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

      4. div-inv 92.0%

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

      5. div-inv 91.9%

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

      6. swap-sqr 92.0%

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

      7. metadata-eval 92.0%

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

      8. pow1/2 92.0%

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

      9. pow-flip 92.1%

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

      10. metadata-eval 92.1%

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

      11. pow1/2 92.1%

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

      12. pow-flip 92.1%

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

      13. metadata-eval 92.1%

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

   9. Applied egg-rr 92.1%

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

       1. *-commutative 92.1%

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

       2. pow-prod-up 92.2%

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

       3. metadata-eval 92.2%

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

       4. inv-pow 92.2%

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

       5. associate-/r/ 92.2%

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

       6. div-inv 92.3%

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

       7. metadata-eval 92.3%

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

   11. Applied egg-rr 92.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+68} \lor \neg \left(y \leq 1.6 \cdot 10^{+118}\right):\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+74} \lor \neg \left(y \leq 1.6 \cdot 10^{+118}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.2e+74) (not (<= y 1.6e+118)))
   (* y (/ -0.3333333333333333 (sqrt x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e+74) || !(y <= 1.6e+118)) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.2d+74)) .or. (.not. (y <= 1.6d+118))) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e+74) || !(y <= 1.6e+118)) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.2e+74) or not (y <= 1.6e+118):
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.2e+74) || !(y <= 1.6e+118))
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.2e+74) || ~((y <= 1.6e+118)))
		tmp = y * (-0.3333333333333333 / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.2e+74], N[Not[LessEqual[y, 1.6e+118]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.20000000000000043e74 or 1.60000000000000008e118 < y

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 99.2%

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

      10. fmm-def 99.2%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      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 95.1%

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

      1. associate-*r* 95.0%

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

   7. Simplified 95.0%

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

      1. sqrt-div 94.9%

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

      2. metadata-eval 94.9%

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

      3. div-inv 95.0%

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

   9. Applied egg-rr 95.0%

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

   if -6.20000000000000043e74 < y < 1.60000000000000008e118

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 92.2%

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

      1. associate-*r/ 92.2%

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

      2. metadata-eval 92.2%

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

   7. Simplified 92.2%

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

      1. metadata-eval 92.2%

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

      2. add-sqr-sqrt 92.1%

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

      3. frac-times 92.0%

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

      4. div-inv 92.0%

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

      5. div-inv 91.9%

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

      6. swap-sqr 92.0%

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

      7. metadata-eval 92.0%

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

      8. pow1/2 92.0%

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

      9. pow-flip 92.1%

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

      10. metadata-eval 92.1%

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

      11. pow1/2 92.1%

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

      12. pow-flip 92.1%

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

      13. metadata-eval 92.1%

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

   9. Applied egg-rr 92.1%

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

       1. *-commutative 92.1%

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

       2. pow-prod-up 92.2%

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

       3. metadata-eval 92.2%

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

       4. inv-pow 92.2%

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

       5. associate-/r/ 92.2%

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

       6. div-inv 92.3%

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

       7. metadata-eval 92.3%

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

   11. Applied egg-rr 92.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+74} \lor \neg \left(y \leq 1.6 \cdot 10^{+118}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.66 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+118}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.66e+74)
   (* y (* -0.3333333333333333 (pow x -0.5)))
   (if (<= y 1.6e+118)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (/ -0.3333333333333333 (/ (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.66e+74) {
		tmp = y * (-0.3333333333333333 * pow(x, -0.5));
	} else if (y <= 1.6e+118) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.66d+74)) then
        tmp = y * ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    else if (y <= 1.6d+118) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.66e+74) {
		tmp = y * (-0.3333333333333333 * Math.pow(x, -0.5));
	} else if (y <= 1.6e+118) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.66e+74:
		tmp = y * (-0.3333333333333333 * math.pow(x, -0.5))
	elif y <= 1.6e+118:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.66e+74)
		tmp = Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5)));
	elseif (y <= 1.6e+118)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.66e+74)
		tmp = y * (-0.3333333333333333 * (x ^ -0.5));
	elseif (y <= 1.6e+118)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.66e+74], N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+118], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.66000000000000001e74

   1. Initial program 99.2%

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

      1. associate--l- 99.2%

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

      2. sub-neg 99.2%

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

      3. +-commutative 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. distribute-frac-neg 99.2%

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

      6. sub-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-/l* 99.1%

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

      10. fmm-def 99.1%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      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 89.8%

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

      1. associate-*r* 89.8%

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

   7. Simplified 89.8%

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

      1. *-un-lft-identity 89.8%

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

      2. inv-pow 89.8%

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

      3. sqrt-pow1 89.8%

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

      4. metadata-eval 89.8%

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

   9. Applied egg-rr 89.8%

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

       1. *-lft-identity 89.8%

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

   11. Simplified 89.8%

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

   if -1.66000000000000001e74 < y < 1.60000000000000008e118

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 92.2%

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

      1. associate-*r/ 92.2%

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

      2. metadata-eval 92.2%

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

   7. Simplified 92.2%

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

      1. metadata-eval 92.2%

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

      2. add-sqr-sqrt 92.1%

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

      3. frac-times 92.0%

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

      4. div-inv 92.0%

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

      5. div-inv 91.9%

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

      6. swap-sqr 92.0%

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

      7. metadata-eval 92.0%

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

      8. pow1/2 92.0%

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

      9. pow-flip 92.1%

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

      10. metadata-eval 92.1%

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

      11. pow1/2 92.1%

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

      12. pow-flip 92.1%

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

      13. metadata-eval 92.1%

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

   9. Applied egg-rr 92.1%

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

       1. *-commutative 92.1%

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

       2. pow-prod-up 92.2%

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

       3. metadata-eval 92.2%

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

       4. inv-pow 92.2%

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

       5. associate-/r/ 92.2%

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

       6. div-inv 92.3%

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

       7. metadata-eval 92.3%

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

   11. Applied egg-rr 92.3%

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

   if 1.60000000000000008e118 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.3%

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

      10. fmm-def 99.3%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      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 99.5%

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

      1. associate-*r* 99.4%

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

   7. Simplified 99.4%

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

      1. sqrt-div 99.4%

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

      2. metadata-eval 99.4%

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

      3. div-inv 99.4%

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

      4. associate-/r/ 99.6%

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

   9. Applied egg-rr 99.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.66 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+118}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+70}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+118}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.3e+70)
   (* y (/ -0.3333333333333333 (sqrt x)))
   (if (<= y 1.6e+118)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (/ -0.3333333333333333 (/ (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.3e+70) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else if (y <= 1.6e+118) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.3d+70)) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else if (y <= 1.6d+118) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.3e+70) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else if (y <= 1.6e+118) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.3e+70:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	elif y <= 1.6e+118:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.3e+70)
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	elseif (y <= 1.6e+118)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.3e+70)
		tmp = y * (-0.3333333333333333 / sqrt(x));
	elseif (y <= 1.6e+118)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.3e+70], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+118], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.3000000000000001e70

   1. Initial program 99.2%

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

      1. associate--l- 99.2%

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

      2. sub-neg 99.2%

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

      3. +-commutative 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. distribute-frac-neg 99.2%

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

      6. sub-neg 99.2%

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

      7. neg-mul-1 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-/l* 99.1%

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

      10. fmm-def 99.1%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      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 89.8%

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

      1. associate-*r* 89.8%

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

   7. Simplified 89.8%

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

      1. sqrt-div 89.5%

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

      2. metadata-eval 89.5%

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

      3. div-inv 89.8%

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

   9. Applied egg-rr 89.8%

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

   if -4.3000000000000001e70 < y < 1.60000000000000008e118

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 92.2%

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

      1. associate-*r/ 92.2%

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

      2. metadata-eval 92.2%

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

   7. Simplified 92.2%

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

      1. metadata-eval 92.2%

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

      2. add-sqr-sqrt 92.1%

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

      3. frac-times 92.0%

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

      4. div-inv 92.0%

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

      5. div-inv 91.9%

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

      6. swap-sqr 92.0%

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

      7. metadata-eval 92.0%

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

      8. pow1/2 92.0%

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

      9. pow-flip 92.1%

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

      10. metadata-eval 92.1%

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

      11. pow1/2 92.1%

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

      12. pow-flip 92.1%

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

      13. metadata-eval 92.1%

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

   9. Applied egg-rr 92.1%

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

       1. *-commutative 92.1%

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

       2. pow-prod-up 92.2%

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

       3. metadata-eval 92.2%

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

       4. inv-pow 92.2%

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

       5. associate-/r/ 92.2%

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

       6. div-inv 92.3%

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

       7. metadata-eval 92.3%

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

   11. Applied egg-rr 92.3%

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

   if 1.60000000000000008e118 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.3%

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

      10. fmm-def 99.3%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      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 99.5%

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

      1. associate-*r* 99.4%

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

   7. Simplified 99.4%

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

      1. sqrt-div 99.4%

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

      2. metadata-eval 99.4%

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

      3. div-inv 99.4%

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

      4. associate-/r/ 99.6%

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

   9. Applied egg-rr 99.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+70}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+118}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4500000:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4500000.0)
   (/ (- (* -0.3333333333333333 (* y (sqrt x))) 0.1111111111111111) x)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4500000.0) {
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	} 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 (x <= 4500000.0d0) then
        tmp = (((-0.3333333333333333d0) * (y * sqrt(x))) - 0.1111111111111111d0) / x
    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 (x <= 4500000.0) {
		tmp = ((-0.3333333333333333 * (y * Math.sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4500000.0:
		tmp = ((-0.3333333333333333 * (y * math.sqrt(x))) - 0.1111111111111111) / x
	else:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4500000.0)
		tmp = Float64(Float64(Float64(-0.3333333333333333 * Float64(y * sqrt(x))) - 0.1111111111111111) / x);
	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 (x <= 4500000.0)
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	else
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4500000.0], N[(N[(N[(-0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5e6

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      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-/l* 99.6%

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

      10. fmm-def 99.6%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      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 x around 0 99.0%

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

   if 4.5e6 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 99.4%

      \[\leadsto \color{blue}{1} + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4500000:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.5%

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

   4. metadata-eval 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. neg-mul-1 99.5%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l/ 99.5%

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

   3. associate-*r/ 99.6%

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

   4. frac-2neg 99.6%

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

   5. associate-*r/ 99.5%

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

   6. metadata-eval 99.5%

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

   7. metadata-eval 99.5%

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

   8. div-inv 99.6%

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

   9. distribute-neg-frac2 99.6%

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

   10. associate-/r* 99.5%

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

   11. clear-num 99.5%

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

   12. distribute-neg-frac 99.5%

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

   13. metadata-eval 99.5%

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

   14. *-commutative 99.5%

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

   15. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 13: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.5%

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

   4. metadata-eval 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. neg-mul-1 99.5%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 14: 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 99.6%

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

   1. sub-neg 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r* 99.5%

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

   4. metadata-eval 99.5%

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

   5. distribute-frac-neg 99.5%

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

   6. neg-mul-1 99.5%

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

   7. times-frac 99.6%

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

   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. Add Preprocessing

Alternative 15: 67.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 + 0.1111111111111111 \cdot \frac{-1}{x}}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3e+107)
   (/
    (- 1.0 (* (/ -0.1111111111111111 x) (/ -0.1111111111111111 x)))
    (+ 1.0 (* 0.1111111111111111 (/ -1.0 x))))
   (if (<= y 2.25e+99)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (/
      (- 1.0 (/ (/ 0.012345679012345678 x) x))
      (- 1.0 (/ -0.1111111111111111 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3e+107) {
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 + (0.1111111111111111 * (-1.0 / x)));
	} else if (y <= 2.25e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3d+107)) then
        tmp = (1.0d0 - (((-0.1111111111111111d0) / x) * ((-0.1111111111111111d0) / x))) / (1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x)))
    else if (y <= 2.25d+99) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (1.0d0 - ((0.012345679012345678d0 / x) / x)) / (1.0d0 - ((-0.1111111111111111d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3e+107) {
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 + (0.1111111111111111 * (-1.0 / x)));
	} else if (y <= 2.25e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3e+107:
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 + (0.1111111111111111 * (-1.0 / x)))
	elif y <= 2.25e+99:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3e+107)
		tmp = Float64(Float64(1.0 - Float64(Float64(-0.1111111111111111 / x) * Float64(-0.1111111111111111 / x))) / Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x))));
	elseif (y <= 2.25e+99)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.012345679012345678 / x) / x)) / Float64(1.0 - Float64(-0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3e+107)
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (1.0 + (0.1111111111111111 * (-1.0 / x)));
	elseif (y <= 2.25e+99)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3e+107], N[(N[(1.0 - N[(N[(-0.1111111111111111 / x), $MachinePrecision] * N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e+99], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.00000000000000023e107

   1. Initial program 99.1%

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

      1. associate--l- 99.1%

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

      2. sub-neg 99.1%

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

      3. +-commutative 99.1%

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

      4. distribute-neg-in 99.1%

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

      5. distribute-frac-neg 99.1%

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

      6. sub-neg 99.1%

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

      7. neg-mul-1 99.1%

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

      8. *-commutative 99.1%

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

      9. associate-/l* 99.1%

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

      10. fmm-def 99.1%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      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 0 4.4%

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

      1. associate-*r/ 4.4%

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

      2. metadata-eval 4.4%

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

   7. Simplified 4.4%

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

      1. metadata-eval 4.4%

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

      2. add-sqr-sqrt 4.4%

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

      3. frac-times 4.4%

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

      4. div-inv 4.4%

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

      5. div-inv 4.4%

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

      6. swap-sqr 4.4%

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

      7. metadata-eval 4.4%

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

      8. pow1/2 4.4%

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

      9. pow-flip 4.4%

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

      10. metadata-eval 4.4%

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

      11. pow1/2 4.4%

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

      12. pow-flip 4.4%

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

      13. metadata-eval 4.4%

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

   9. Applied egg-rr 4.4%

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

       1. sub-neg 4.4%

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

       2. flip-+ 4.3%

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

   11. Applied egg-rr 4.3%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-prod 5.5%

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

       3. frac-times 5.5%

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

       4. sqrt-div 5.5%

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

       5. metadata-eval 5.5%

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

       6. metadata-eval 5.5%

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

       7. sqrt-unprod 17.3%

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

       8. add-sqr-sqrt 17.3%

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

       9. div-inv 17.3%

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

   13. Applied egg-rr 17.3%

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

   if -3.00000000000000023e107 < y < 2.25e99

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. *-commutative 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      9. associate-/l* 99.8%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 99.7%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      11. associate-/r* 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      17. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 90.1%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 90.1%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 90.1%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 90.1%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. metadata-eval 90.1%

         \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

      2. add-sqr-sqrt 89.9%

         \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      3. frac-times 89.9%

         \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

      4. div-inv 89.9%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

      5. div-inv 89.8%

         \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

      6. swap-sqr 89.9%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

      7. metadata-eval 89.9%

         \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

      8. pow1/2 89.9%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

      9. pow-flip 90.0%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

      10. metadata-eval 90.0%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

      11. pow1/2 90.0%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      12. pow-flip 90.0%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      13. metadata-eval 90.0%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

   9. Applied egg-rr 90.0%

      \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 90.0%

          \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot 0.1111111111111111} \]

       2. pow-prod-up 90.1%

          \[\leadsto 1 - \color{blue}{{x}^{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

       3. metadata-eval 90.1%

          \[\leadsto 1 - {x}^{\color{blue}{-1}} \cdot 0.1111111111111111 \]

       4. inv-pow 90.1%

          \[\leadsto 1 - \color{blue}{\frac{1}{x}} \cdot 0.1111111111111111 \]

       5. associate-/r/ 90.1%

          \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

       6. div-inv 90.2%

          \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

       7. metadata-eval 90.2%

          \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

   11. Applied egg-rr 90.2%

       \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]

   if 2.25e99 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.6%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.6%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.6%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      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-/l* 99.4%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 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* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      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 0 6.4%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 6.4%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 6.4%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. metadata-eval 6.4%

         \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

      2. add-sqr-sqrt 6.4%

         \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      3. frac-times 6.4%

         \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

      4. div-inv 6.4%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

      5. div-inv 6.4%

         \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

      6. swap-sqr 6.4%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

      7. metadata-eval 6.4%

         \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

      8. pow1/2 6.4%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

      9. pow-flip 6.4%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

      10. metadata-eval 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

      11. pow1/2 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      12. pow-flip 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      13. metadata-eval 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

   9. Applied egg-rr 6.4%

      \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. sub-neg 6.4%

          \[\leadsto \color{blue}{1 + \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)} \]

       2. flip-+ 26.1%

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right) \cdot \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)}{1 - \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)}} \]

   11. Applied egg-rr 26.1%

       \[\leadsto \color{blue}{\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - \frac{-0.1111111111111111}{x}}} \]
   12. Step-by-step derivation

       1. frac-times 26.1%

          \[\leadsto \frac{1 - \color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       2. associate-/r* 26.1%

          \[\leadsto \frac{1 - \color{blue}{\frac{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x}}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       3. metadata-eval 26.1%

          \[\leadsto \frac{1 - \frac{\frac{\color{blue}{0.012345679012345678}}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}} \]

   13. Applied egg-rr 26.1%

       \[\leadsto \frac{1 - \color{blue}{\frac{\frac{0.012345679012345678}{x}}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 + 0.1111111111111111 \cdot \frac{-1}{x}}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.012345679012345678}{x}}{x}\\ t_1 := 1 - \frac{-0.1111111111111111}{x}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + t\_0}{t\_1}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ 0.012345679012345678 x) x))
        (t_1 (- 1.0 (/ -0.1111111111111111 x))))
   (if (<= y -3e+107)
     (/ (+ 1.0 t_0) t_1)
     (if (<= y 2.25e+99) (+ 1.0 (/ -1.0 (* x 9.0))) (/ (- 1.0 t_0) t_1)))))

C

double code(double x, double y) {
	double t_0 = (0.012345679012345678 / x) / x;
	double t_1 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -3e+107) {
		tmp = (1.0 + t_0) / t_1;
	} else if (y <= 2.25e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 - t_0) / t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (0.012345679012345678d0 / x) / x
    t_1 = 1.0d0 - ((-0.1111111111111111d0) / x)
    if (y <= (-3d+107)) then
        tmp = (1.0d0 + t_0) / t_1
    else if (y <= 2.25d+99) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (1.0d0 - t_0) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (0.012345679012345678 / x) / x;
	double t_1 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -3e+107) {
		tmp = (1.0 + t_0) / t_1;
	} else if (y <= 2.25e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 - t_0) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (0.012345679012345678 / x) / x
	t_1 = 1.0 - (-0.1111111111111111 / x)
	tmp = 0
	if y <= -3e+107:
		tmp = (1.0 + t_0) / t_1
	elif y <= 2.25e+99:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (1.0 - t_0) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(0.012345679012345678 / x) / x)
	t_1 = Float64(1.0 - Float64(-0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -3e+107)
		tmp = Float64(Float64(1.0 + t_0) / t_1);
	elseif (y <= 2.25e+99)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(1.0 - t_0) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (0.012345679012345678 / x) / x;
	t_1 = 1.0 - (-0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= -3e+107)
		tmp = (1.0 + t_0) / t_1;
	elseif (y <= 2.25e+99)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (1.0 - t_0) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e+107], N[(N[(1.0 + t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 2.25e+99], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.00000000000000023e107

   1. Initial program 99.1%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.1%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.1%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.1%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.1%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.1%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. *-commutative 99.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      9. associate-/l* 99.1%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 99.1%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      11. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      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 0 4.4%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 4.4%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 4.4%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 4.4%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. metadata-eval 4.4%

         \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

      2. add-sqr-sqrt 4.4%

         \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      3. frac-times 4.4%

         \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

      4. div-inv 4.4%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

      5. div-inv 4.4%

         \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

      6. swap-sqr 4.4%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

      7. metadata-eval 4.4%

         \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

      8. pow1/2 4.4%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

      9. pow-flip 4.4%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

      10. metadata-eval 4.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

      11. pow1/2 4.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      12. pow-flip 4.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      13. metadata-eval 4.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

   9. Applied egg-rr 4.4%

      \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. sub-neg 4.4%

          \[\leadsto \color{blue}{1 + \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)} \]

       2. flip-+ 4.3%

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right) \cdot \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)}{1 - \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)}} \]

   11. Applied egg-rr 4.3%

       \[\leadsto \color{blue}{\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - \frac{-0.1111111111111111}{x}}} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \color{blue}{\left(\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}\right)}}{1 - \frac{-0.1111111111111111}{x}} \]

       2. sqrt-prod 17.3%

          \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}}}{1 - \frac{-0.1111111111111111}{x}} \]

       3. frac-times 17.3%

          \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}}}{1 - \frac{-0.1111111111111111}{x}} \]

       4. sqrt-div 17.3%

          \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \color{blue}{\frac{\sqrt{-0.1111111111111111 \cdot -0.1111111111111111}}{\sqrt{x \cdot x}}}}{1 - \frac{-0.1111111111111111}{x}} \]

       5. metadata-eval 17.3%

          \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{\sqrt{\color{blue}{0.012345679012345678}}}{\sqrt{x \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       6. metadata-eval 17.3%

          \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{\color{blue}{0.1111111111111111}}{\sqrt{x \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       7. sqrt-unprod 17.3%

          \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{1 - \frac{-0.1111111111111111}{x}} \]

       8. metadata-eval 17.3%

          \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{\color{blue}{--0.1111111111111111}}{\sqrt{x} \cdot \sqrt{x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       9. add-sqr-sqrt 17.3%

          \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{--0.1111111111111111}{\color{blue}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       10. distribute-neg-frac 17.3%

           \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)}}{1 - \frac{-0.1111111111111111}{x}} \]

       11. distribute-rgt-neg-in 17.3%

           \[\leadsto \frac{1 - \color{blue}{\left(-\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}\right)}}{1 - \frac{-0.1111111111111111}{x}} \]

       12. associate-*l/ 17.3%

           \[\leadsto \frac{1 - \left(-\color{blue}{\frac{-0.1111111111111111 \cdot \frac{-0.1111111111111111}{x}}{x}}\right)}{1 - \frac{-0.1111111111111111}{x}} \]

       13. distribute-neg-frac2 17.3%

           \[\leadsto \frac{1 - \color{blue}{\frac{-0.1111111111111111 \cdot \frac{-0.1111111111111111}{x}}{-x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       14. associate-*r/ 17.3%

           \[\leadsto \frac{1 - \frac{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x}}}{-x}}{1 - \frac{-0.1111111111111111}{x}} \]

       15. metadata-eval 17.3%

           \[\leadsto \frac{1 - \frac{\frac{\color{blue}{0.012345679012345678}}{x}}{-x}}{1 - \frac{-0.1111111111111111}{x}} \]

   13. Applied egg-rr 17.3%

       \[\leadsto \frac{1 - \color{blue}{\frac{\frac{0.012345679012345678}{x}}{-x}}}{1 - \frac{-0.1111111111111111}{x}} \]

   if -3.00000000000000023e107 < y < 2.25e99

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.8%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.8%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.8%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.8%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.8%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.8%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. *-commutative 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      9. associate-/l* 99.8%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 99.7%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      11. associate-/r* 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.8%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      17. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 90.1%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 90.1%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 90.1%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 90.1%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. metadata-eval 90.1%

         \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

      2. add-sqr-sqrt 89.9%

         \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      3. frac-times 89.9%

         \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

      4. div-inv 89.9%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

      5. div-inv 89.8%

         \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

      6. swap-sqr 89.9%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

      7. metadata-eval 89.9%

         \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

      8. pow1/2 89.9%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

      9. pow-flip 90.0%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

      10. metadata-eval 90.0%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

      11. pow1/2 90.0%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      12. pow-flip 90.0%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      13. metadata-eval 90.0%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

   9. Applied egg-rr 90.0%

      \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 90.0%

          \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot 0.1111111111111111} \]

       2. pow-prod-up 90.1%

          \[\leadsto 1 - \color{blue}{{x}^{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

       3. metadata-eval 90.1%

          \[\leadsto 1 - {x}^{\color{blue}{-1}} \cdot 0.1111111111111111 \]

       4. inv-pow 90.1%

          \[\leadsto 1 - \color{blue}{\frac{1}{x}} \cdot 0.1111111111111111 \]

       5. associate-/r/ 90.1%

          \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

       6. div-inv 90.2%

          \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

       7. metadata-eval 90.2%

          \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

   11. Applied egg-rr 90.2%

       \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]

   if 2.25e99 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.6%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.6%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.6%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      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-/l* 99.4%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 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* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      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 0 6.4%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 6.4%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 6.4%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. metadata-eval 6.4%

         \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

      2. add-sqr-sqrt 6.4%

         \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      3. frac-times 6.4%

         \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

      4. div-inv 6.4%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

      5. div-inv 6.4%

         \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

      6. swap-sqr 6.4%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

      7. metadata-eval 6.4%

         \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

      8. pow1/2 6.4%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

      9. pow-flip 6.4%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

      10. metadata-eval 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

      11. pow1/2 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      12. pow-flip 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      13. metadata-eval 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

   9. Applied egg-rr 6.4%

      \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. sub-neg 6.4%

          \[\leadsto \color{blue}{1 + \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)} \]

       2. flip-+ 26.1%

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right) \cdot \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)}{1 - \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)}} \]

   11. Applied egg-rr 26.1%

       \[\leadsto \color{blue}{\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - \frac{-0.1111111111111111}{x}}} \]
   12. Step-by-step derivation

       1. frac-times 26.1%

          \[\leadsto \frac{1 - \color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       2. associate-/r* 26.1%

          \[\leadsto \frac{1 - \color{blue}{\frac{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x}}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       3. metadata-eval 26.1%

          \[\leadsto \frac{1 - \frac{\frac{\color{blue}{0.012345679012345678}}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}} \]

   13. Applied egg-rr 26.1%

       \[\leadsto \frac{1 - \color{blue}{\frac{\frac{0.012345679012345678}{x}}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.25e+99)
   (+ 1.0 (/ -1.0 (* x 9.0)))
   (/
    (- 1.0 (/ (/ 0.012345679012345678 x) x))
    (- 1.0 (/ -0.1111111111111111 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.25e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.25d+99) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (1.0d0 - ((0.012345679012345678d0 / x) / x)) / (1.0d0 - ((-0.1111111111111111d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.25e+99) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.25e+99:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.25e+99)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.012345679012345678 / x) / x)) / Float64(1.0 - Float64(-0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.25e+99)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.25e+99], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e99

   1. Initial program 99.7%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.7%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.7%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.7%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.7%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.7%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.7%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.7%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. *-commutative 99.7%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      9. associate-/l* 99.6%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 99.6%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      11. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      17. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 74.5%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 74.5%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 74.5%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 74.5%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. metadata-eval 74.5%

         \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

      2. add-sqr-sqrt 74.4%

         \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      3. frac-times 74.4%

         \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

      4. div-inv 74.4%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

      5. div-inv 74.3%

         \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

      6. swap-sqr 74.4%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

      7. metadata-eval 74.4%

         \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

      8. pow1/2 74.4%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

      9. pow-flip 74.5%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

      10. metadata-eval 74.5%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

      11. pow1/2 74.5%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      12. pow-flip 74.5%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      13. metadata-eval 74.5%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

   9. Applied egg-rr 74.5%

      \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 74.5%

          \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot 0.1111111111111111} \]

       2. pow-prod-up 74.5%

          \[\leadsto 1 - \color{blue}{{x}^{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

       3. metadata-eval 74.5%

          \[\leadsto 1 - {x}^{\color{blue}{-1}} \cdot 0.1111111111111111 \]

       4. inv-pow 74.5%

          \[\leadsto 1 - \color{blue}{\frac{1}{x}} \cdot 0.1111111111111111 \]

       5. associate-/r/ 74.6%

          \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

       6. div-inv 74.6%

          \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

       7. metadata-eval 74.6%

          \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

   11. Applied egg-rr 74.6%

       \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]

   if 2.25e99 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.6%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.6%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.6%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      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-/l* 99.4%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 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* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      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 0 6.4%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 6.4%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 6.4%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. metadata-eval 6.4%

         \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

      2. add-sqr-sqrt 6.4%

         \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      3. frac-times 6.4%

         \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

      4. div-inv 6.4%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

      5. div-inv 6.4%

         \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

      6. swap-sqr 6.4%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

      7. metadata-eval 6.4%

         \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

      8. pow1/2 6.4%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

      9. pow-flip 6.4%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

      10. metadata-eval 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

      11. pow1/2 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      12. pow-flip 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      13. metadata-eval 6.4%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

   9. Applied egg-rr 6.4%

      \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. sub-neg 6.4%

          \[\leadsto \color{blue}{1 + \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)} \]

       2. flip-+ 26.1%

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right) \cdot \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)}{1 - \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)}} \]

   11. Applied egg-rr 26.1%

       \[\leadsto \color{blue}{\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - \frac{-0.1111111111111111}{x}}} \]
   12. Step-by-step derivation

       1. frac-times 26.1%

          \[\leadsto \frac{1 - \color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       2. associate-/r* 26.1%

          \[\leadsto \frac{1 - \color{blue}{\frac{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x}}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       3. metadata-eval 26.1%

          \[\leadsto \frac{1 - \frac{\frac{\color{blue}{0.012345679012345678}}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}} \]

   13. Applied egg-rr 26.1%

       \[\leadsto \frac{1 - \color{blue}{\frac{\frac{0.012345679012345678}{x}}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 66.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+171}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{\frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.5e+171)
   (+ 1.0 (/ -1.0 (* x 9.0)))
   (/
    (- 1.0 (* (/ -0.1111111111111111 x) (/ -0.1111111111111111 x)))
    (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.5e+171) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.5d+171) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (1.0d0 - (((-0.1111111111111111d0) / x) * ((-0.1111111111111111d0) / x))) / (0.1111111111111111d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.5e+171) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.5e+171:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.5e+171)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(-0.1111111111111111 / x) * Float64(-0.1111111111111111 / x))) / Float64(0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.5e+171)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (1.0 - ((-0.1111111111111111 / x) * (-0.1111111111111111 / x))) / (0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.5e+171], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(-0.1111111111111111 / x), $MachinePrecision] * N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5e171

   1. Initial program 99.7%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.7%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.7%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.7%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.7%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.7%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.7%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.7%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. *-commutative 99.7%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      9. associate-/l* 99.6%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 99.6%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      11. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      17. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 72.1%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 72.2%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 72.2%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 72.2%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. metadata-eval 72.2%

         \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

      2. add-sqr-sqrt 72.1%

         \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      3. frac-times 72.0%

         \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

      4. div-inv 72.0%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

      5. div-inv 71.9%

         \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

      6. swap-sqr 72.0%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

      7. metadata-eval 72.0%

         \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

      8. pow1/2 72.0%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

      9. pow-flip 72.1%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

      10. metadata-eval 72.1%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

      11. pow1/2 72.1%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      12. pow-flip 72.1%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      13. metadata-eval 72.1%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

   9. Applied egg-rr 72.1%

      \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 72.1%

          \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot 0.1111111111111111} \]

       2. pow-prod-up 72.1%

          \[\leadsto 1 - \color{blue}{{x}^{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

       3. metadata-eval 72.1%

          \[\leadsto 1 - {x}^{\color{blue}{-1}} \cdot 0.1111111111111111 \]

       4. inv-pow 72.1%

          \[\leadsto 1 - \color{blue}{\frac{1}{x}} \cdot 0.1111111111111111 \]

       5. associate-/r/ 72.2%

          \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

       6. div-inv 72.3%

          \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

       7. metadata-eval 72.3%

          \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

   11. Applied egg-rr 72.3%

       \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]

   if 6.5e171 < y

   1. Initial program 99.5%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.5%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.5%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.5%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.5%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.5%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.5%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.5%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. *-commutative 99.5%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      9. associate-/l* 99.4%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 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* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      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 0 3.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 3.9%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 3.9%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 3.9%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. metadata-eval 3.9%

         \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

      2. add-sqr-sqrt 3.9%

         \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      3. frac-times 3.9%

         \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

      4. div-inv 3.9%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

      5. div-inv 3.9%

         \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

      6. swap-sqr 3.9%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

      7. metadata-eval 3.9%

         \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

      8. pow1/2 3.9%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

      9. pow-flip 3.9%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

      10. metadata-eval 3.9%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

      11. pow1/2 3.9%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      12. pow-flip 3.9%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      13. metadata-eval 3.9%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

   9. Applied egg-rr 3.9%

      \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]
   10. Step-by-step derivation

       1. sub-neg 3.9%

          \[\leadsto \color{blue}{1 + \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)} \]

       2. flip-+ 27.7%

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right) \cdot \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)}{1 - \left(-0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right)}} \]

   11. Applied egg-rr 27.7%

       \[\leadsto \color{blue}{\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 - \frac{-0.1111111111111111}{x}}} \]

   12. Taylor expanded in x around 0 27.5%

       \[\leadsto \frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{\color{blue}{\frac{0.1111111111111111}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+171}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{\frac{0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.0% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.6%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.6%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.6%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      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-/l* 99.6%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 99.6%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      11. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      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 0 57.2%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 57.3%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 57.3%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 57.3%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
   8. Step-by-step derivation

      1. metadata-eval 57.3%

         \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

      2. add-sqr-sqrt 57.1%

         \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      3. frac-times 57.0%

         \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

      4. div-inv 57.0%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

      5. div-inv 56.9%

         \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

      6. swap-sqr 57.0%

         \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

      7. metadata-eval 57.0%

         \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

      8. pow1/2 57.0%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

      9. pow-flip 57.1%

         \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

      10. metadata-eval 57.1%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

      11. pow1/2 57.1%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      12. pow-flip 57.1%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      13. metadata-eval 57.1%

          \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

   9. Applied egg-rr 57.1%

      \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]

   10. Taylor expanded in x around 0 57.0%

       \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} \]

   if 0.112000000000000002 < x

   1. Initial program 99.7%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.7%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. sub-neg 99.7%

         \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

      3. +-commutative 99.7%

         \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

      4. distribute-neg-in 99.7%

         \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

      5. distribute-frac-neg 99.7%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

      6. sub-neg 99.7%

         \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.7%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. *-commutative 99.7%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      9. associate-/l* 99.6%

         \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

      10. fmm-def 99.6%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

      11. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

      12. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      17. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 64.2%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 64.2%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

      2. metadata-eval 64.2%

         \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]

   8. Taylor expanded in x around inf 63.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 64.2% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ -1.0 (* x 9.0))))

C

double code(double x, double y) {
	return 1.0 + (-1.0 / (x * 9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-1.0 / (x * 9.0));
}

Python

def code(x, y):
	return 1.0 + (-1.0 / (x * 9.0))

Julia

function code(x, y)
	return Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-1.0 / (x * 9.0));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.6%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. sub-neg 99.6%

      \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

   3. +-commutative 99.6%

      \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

   4. distribute-neg-in 99.6%

      \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

   5. distribute-frac-neg 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   6. sub-neg 99.6%

      \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   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-/l* 99.6%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fmm-def 99.6%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   11. associate-/r* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   12. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   17. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 60.7%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. associate-*r/ 60.7%

      \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

   2. metadata-eval 60.7%

      \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

7. Simplified 60.7%

   \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
8. Step-by-step derivation

   1. metadata-eval 60.7%

      \[\leadsto 1 - \frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{x} \]

   2. add-sqr-sqrt 60.6%

      \[\leadsto 1 - \frac{-0.3333333333333333 \cdot -0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

   3. frac-times 60.5%

      \[\leadsto 1 - \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

   4. div-inv 60.5%

      \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \frac{-0.3333333333333333}{\sqrt{x}} \]

   5. div-inv 60.5%

      \[\leadsto 1 - \left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\right)} \]

   6. swap-sqr 60.6%

      \[\leadsto 1 - \color{blue}{\left(-0.3333333333333333 \cdot -0.3333333333333333\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right)} \]

   7. metadata-eval 60.6%

      \[\leadsto 1 - \color{blue}{0.1111111111111111} \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}\right) \]

   8. pow1/2 60.6%

      \[\leadsto 1 - 0.1111111111111111 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\sqrt{x}}\right) \]

   9. pow-flip 60.6%

      \[\leadsto 1 - 0.1111111111111111 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\sqrt{x}}\right) \]

   10. metadata-eval 60.6%

       \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{\color{blue}{-0.5}} \cdot \frac{1}{\sqrt{x}}\right) \]

   11. pow1/2 60.6%

       \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

   12. pow-flip 60.6%

       \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

   13. metadata-eval 60.6%

       \[\leadsto 1 - 0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{\color{blue}{-0.5}}\right) \]

9. Applied egg-rr 60.6%

   \[\leadsto 1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \]
10. Step-by-step derivation

    1. *-commutative 60.6%

       \[\leadsto 1 - \color{blue}{\left({x}^{-0.5} \cdot {x}^{-0.5}\right) \cdot 0.1111111111111111} \]

    2. pow-prod-up 60.7%

       \[\leadsto 1 - \color{blue}{{x}^{\left(-0.5 + -0.5\right)}} \cdot 0.1111111111111111 \]

    3. metadata-eval 60.7%

       \[\leadsto 1 - {x}^{\color{blue}{-1}} \cdot 0.1111111111111111 \]

    4. inv-pow 60.7%

       \[\leadsto 1 - \color{blue}{\frac{1}{x}} \cdot 0.1111111111111111 \]

    5. associate-/r/ 60.7%

       \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

    6. div-inv 60.8%

       \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

    7. metadata-eval 60.8%

       \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

11. Applied egg-rr 60.8%

    \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]

12. Final simplification 60.8%

    \[\leadsto 1 + \frac{-1}{x \cdot 9} \]
13. Add Preprocessing

Alternative 21: 64.1% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ 0.1111111111111111 x)))

C

double code(double x, double y) {
	return 1.0 - (0.1111111111111111 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (0.1111111111111111d0 / x)
end function

Java

public static double code(double x, double y) {
	return 1.0 - (0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 - (0.1111111111111111 / x)

Julia

function code(x, y)
	return Float64(1.0 - Float64(0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - (0.1111111111111111 / x);
end

Wolfram

code[x_, y_] := N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.6%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. sub-neg 99.6%

      \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

   3. +-commutative 99.6%

      \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

   4. distribute-neg-in 99.6%

      \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

   5. distribute-frac-neg 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   6. sub-neg 99.6%

      \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   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-/l* 99.6%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fmm-def 99.6%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   11. associate-/r* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   12. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   17. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 60.7%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. associate-*r/ 60.7%

      \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

   2. metadata-eval 60.7%

      \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

7. Simplified 60.7%

   \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
8. Add Preprocessing

Alternative 22: 32.3% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. associate--l- 99.6%

      \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. sub-neg 99.6%

      \[\leadsto \color{blue}{1 + \left(-\left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)\right)} \]

   3. +-commutative 99.6%

      \[\leadsto 1 + \left(-\color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)}\right) \]

   4. distribute-neg-in 99.6%

      \[\leadsto 1 + \color{blue}{\left(\left(-\frac{y}{3 \cdot \sqrt{x}}\right) + \left(-\frac{1}{x \cdot 9}\right)\right)} \]

   5. distribute-frac-neg 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} + \left(-\frac{1}{x \cdot 9}\right)\right) \]

   6. sub-neg 99.6%

      \[\leadsto 1 + \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   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-/l* 99.6%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fmm-def 99.6%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   11. associate-/r* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   12. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   17. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 60.7%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. associate-*r/ 60.7%

      \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

   2. metadata-eval 60.7%

      \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

7. Simplified 60.7%

   \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]

8. Taylor expanded in x around inf 32.2%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 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]

Reproduce

herbie shell --seed 2024172 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x)))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))