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

Percentage Accurate: 99.7% → 99.7%

Time: 8.4s

Alternatives: 9

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 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+83}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+38}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{{x}^{-0.5}}{\frac{-3}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e+83)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 9e+38)
     (- 1.0 (pow (* x 9.0) -1.0))
     (+ 1.0 (/ (pow x -0.5) (/ -3.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+83) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 9e+38) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (pow(x, -0.5) / (-3.0 / 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 <= (-1.1d+83)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 9d+38) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 + ((x ** (-0.5d0)) / ((-3.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+83) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 9e+38) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (Math.pow(x, -0.5) / (-3.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e+83:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 9e+38:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 + (math.pow(x, -0.5) / (-3.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e+83)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 9e+38)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64((x ^ -0.5) / Float64(-3.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.1e+83)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 9e+38)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 + ((x ^ -0.5) / (-3.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e+83], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e+38], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Power[x, -0.5], $MachinePrecision] / N[(-3.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.09999999999999999e83

   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* 99.5%

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

      10. fmm-def 99.5%

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

      11. associate-/r* 99.6%

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

      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* 99.6%

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

      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 inf 99.4%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

      1. add-log-exp 15.5%

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

      2. *-un-lft-identity 15.5%

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

      3. log-prod 15.5%

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

      4. metadata-eval 15.5%

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

      5. add-log-exp 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*r* 99.4%

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

      8. sqrt-div 99.6%

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

      9. metadata-eval 99.6%

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

      10. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. +-lft-identity 99.6%

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

   11. Simplified 99.6%

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

   if -1.09999999999999999e83 < y < 8.99999999999999961e38

   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* 99.7%

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

      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 98.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.4%

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

      3. frac-times 43.4%

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

      4. metadata-eval 43.4%

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

      5. metadata-eval 43.4%

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

      6. frac-times 43.4%

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

      7. clear-num 43.4%

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

      8. clear-num 43.4%

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

      9. frac-times 43.4%

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

      10. div-inv 43.4%

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

      11. metadata-eval 43.4%

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

      12. div-inv 43.4%

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

      13. metadata-eval 43.4%

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

      14. frac-times 43.4%

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

      15. sqrt-unprod 43.5%

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

      16. add-sqr-sqrt 43.5%

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

      17. metadata-eval 43.5%

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

      18. div-inv 43.5%

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

      19. frac-2neg 43.5%

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

      20. metadata-eval 43.5%

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

      21. clear-num 43.5%

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

      22. distribute-frac-neg2 43.5%

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

   7. Applied egg-rr 43.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.2%

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

      3. frac-times 71.2%

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

      4. metadata-eval 71.2%

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

      5. metadata-eval 71.2%

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

      6. frac-times 71.2%

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

      7. clear-num 71.2%

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

      8. clear-num 71.2%

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

      9. frac-times 71.1%

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

      10. div-inv 71.2%

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

      11. metadata-eval 71.2%

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

      12. div-inv 71.2%

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

      13. metadata-eval 71.2%

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

      14. frac-times 71.2%

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

      15. sqrt-unprod 97.8%

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

      16. add-sqr-sqrt 98.1%

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

      17. inv-pow 98.1%

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

   9. Applied egg-rr 98.1%

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

   if 8.99999999999999961e38 < y

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. 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* 99.7%

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

      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* 99.6%

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

      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 inf 94.6%

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

      1. associate-*r* 94.6%

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

      2. *-commutative 94.6%

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

      3. associate-*l* 94.6%

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

   7. Simplified 94.6%

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

      1. add-log-exp 27.8%

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

      2. *-un-lft-identity 27.8%

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

      3. log-prod 27.8%

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

      4. metadata-eval 27.8%

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

      5. add-log-exp 94.6%

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

      6. *-commutative 94.6%

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

      7. associate-*r* 94.6%

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

      8. sqrt-div 94.5%

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

      9. metadata-eval 94.5%

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

      10. un-div-inv 94.7%

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

   9. Applied egg-rr 94.7%

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

       1. +-lft-identity 94.7%

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

   11. Simplified 94.7%

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

       1. clear-num 94.5%

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

       2. un-div-inv 94.6%

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

   13. Applied egg-rr 94.6%

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

       1. associate-/r/ 94.6%

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

   15. Simplified 94.6%

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

       1. associate-*l/ 94.6%

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

       2. clear-num 94.5%

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

       3. div-inv 94.4%

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

       4. associate-/r* 94.7%

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

       5. pow1/2 94.7%

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

       6. pow-flip 94.7%

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

       7. metadata-eval 94.7%

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

       8. associate-/r* 94.8%

          \[\leadsto 1 + \frac{{x}^{-0.5}}{\color{blue}{\frac{\frac{1}{-0.3333333333333333}}{y}}} \]

       9. metadata-eval 94.8%

          \[\leadsto 1 + \frac{{x}^{-0.5}}{\frac{\color{blue}{-3}}{y}} \]

   17. Applied egg-rr 94.8%

       \[\leadsto 1 + \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+83}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+38}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{{x}^{-0.5}}{\frac{-3}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+82} \lor \neg \left(y \leq 9.5 \cdot 10^{+37}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.2e+82) (not (<= y 9.5e+37)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.2e+82) || !(y <= 9.5e+37)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-9.2d+82)) .or. (.not. (y <= 9.5d+37))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9.2e+82) || !(y <= 9.5e+37)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.2e+82) or not (y <= 9.5e+37):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.2e+82) || !(y <= 9.5e+37))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9.2e+82) || ~((y <= 9.5e+37)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.2e+82], N[Not[LessEqual[y, 9.5e+37]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.19999999999999953e82 or 9.4999999999999995e37 < 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* 99.6%

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

      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* 99.6%

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

      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 inf 96.8%

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

      1. associate-*r* 96.8%

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

      2. *-commutative 96.8%

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

      3. associate-*l* 96.8%

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

   7. Simplified 96.8%

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

      1. add-log-exp 22.2%

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

      2. *-un-lft-identity 22.2%

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

      3. log-prod 22.2%

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

      4. metadata-eval 22.2%

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

      5. add-log-exp 96.8%

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

      6. *-commutative 96.8%

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

      7. associate-*r* 96.8%

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

      8. sqrt-div 96.8%

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

      9. metadata-eval 96.8%

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

      10. un-div-inv 96.9%

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

   9. Applied egg-rr 96.9%

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

       1. +-lft-identity 96.9%

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

   11. Simplified 96.9%

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

   if -9.19999999999999953e82 < y < 9.4999999999999995e37

   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* 99.7%

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

      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 98.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.4%

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

      3. frac-times 43.4%

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

      4. metadata-eval 43.4%

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

      5. metadata-eval 43.4%

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

      6. frac-times 43.4%

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

      7. clear-num 43.4%

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

      8. clear-num 43.4%

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

      9. frac-times 43.4%

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

      10. div-inv 43.4%

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

      11. metadata-eval 43.4%

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

      12. div-inv 43.4%

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

      13. metadata-eval 43.4%

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

      14. frac-times 43.4%

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

      15. sqrt-unprod 43.5%

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

      16. add-sqr-sqrt 43.5%

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

      17. metadata-eval 43.5%

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

      18. div-inv 43.5%

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

      19. frac-2neg 43.5%

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

      20. metadata-eval 43.5%

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

      21. clear-num 43.5%

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

      22. distribute-frac-neg2 43.5%

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

   7. Applied egg-rr 43.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.2%

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

      3. frac-times 71.2%

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

      4. metadata-eval 71.2%

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

      5. metadata-eval 71.2%

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

      6. frac-times 71.2%

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

      7. clear-num 71.2%

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

      8. clear-num 71.2%

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

      9. frac-times 71.1%

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

      10. div-inv 71.2%

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

      11. metadata-eval 71.2%

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

      12. div-inv 71.2%

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

      13. metadata-eval 71.2%

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

      14. frac-times 71.2%

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

      15. sqrt-unprod 97.8%

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

      16. add-sqr-sqrt 98.1%

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

      17. inv-pow 98.1%

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

   9. Applied egg-rr 98.1%

      \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+82} \lor \neg \left(y \leq 9.5 \cdot 10^{+37}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+82}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+38}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.2e+82)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 2.8e+38)
     (- 1.0 (pow (* x 9.0) -1.0))
     (+ 1.0 (/ y (* (sqrt x) -3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.2e+82) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 2.8e+38) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9.2d+82)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 2.8d+38) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.2e+82) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 2.8e+38) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (y / (Math.sqrt(x) * -3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.2e+82:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 2.8e+38:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 + (y / (math.sqrt(x) * -3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.2e+82)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 2.8e+38)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.2e+82)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 2.8e+38)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.2e+82], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e+38], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.19999999999999953e82

   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* 99.5%

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

      10. fmm-def 99.5%

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

      11. associate-/r* 99.6%

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

      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* 99.6%

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

      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 inf 99.4%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

      1. add-log-exp 15.5%

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

      2. *-un-lft-identity 15.5%

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

      3. log-prod 15.5%

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

      4. metadata-eval 15.5%

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

      5. add-log-exp 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*r* 99.4%

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

      8. sqrt-div 99.6%

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

      9. metadata-eval 99.6%

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

      10. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. +-lft-identity 99.6%

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

   11. Simplified 99.6%

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

   if -9.19999999999999953e82 < y < 2.8e38

   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* 99.7%

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

      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 98.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.4%

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

      3. frac-times 43.4%

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

      4. metadata-eval 43.4%

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

      5. metadata-eval 43.4%

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

      6. frac-times 43.4%

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

      7. clear-num 43.4%

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

      8. clear-num 43.4%

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

      9. frac-times 43.4%

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

      10. div-inv 43.4%

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

      11. metadata-eval 43.4%

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

      12. div-inv 43.4%

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

      13. metadata-eval 43.4%

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

      14. frac-times 43.4%

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

      15. sqrt-unprod 43.5%

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

      16. add-sqr-sqrt 43.5%

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

      17. metadata-eval 43.5%

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

      18. div-inv 43.5%

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

      19. frac-2neg 43.5%

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

      20. metadata-eval 43.5%

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

      21. clear-num 43.5%

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

      22. distribute-frac-neg2 43.5%

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

   7. Applied egg-rr 43.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.2%

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

      3. frac-times 71.2%

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

      4. metadata-eval 71.2%

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

      5. metadata-eval 71.2%

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

      6. frac-times 71.2%

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

      7. clear-num 71.2%

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

      8. clear-num 71.2%

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

      9. frac-times 71.1%

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

      10. div-inv 71.2%

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

      11. metadata-eval 71.2%

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

      12. div-inv 71.2%

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

      13. metadata-eval 71.2%

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

      14. frac-times 71.2%

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

      15. sqrt-unprod 97.8%

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

      16. add-sqr-sqrt 98.1%

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

      17. inv-pow 98.1%

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

   9. Applied egg-rr 98.1%

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

   if 2.8e38 < y

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. 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* 99.7%

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

      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* 99.6%

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

      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 inf 94.6%

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

      1. associate-*r* 94.6%

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

      2. *-commutative 94.6%

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

      3. associate-*l* 94.6%

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

   7. Simplified 94.6%

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

      1. add-log-exp 27.8%

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

      2. *-un-lft-identity 27.8%

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

      3. log-prod 27.8%

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

      4. metadata-eval 27.8%

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

      5. add-log-exp 94.6%

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

      6. *-commutative 94.6%

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

      7. associate-*r* 94.6%

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

      8. sqrt-div 94.5%

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

      9. metadata-eval 94.5%

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

      10. un-div-inv 94.7%

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

   9. Applied egg-rr 94.7%

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

       1. +-lft-identity 94.7%

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

   11. Simplified 94.7%

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

       1. *-commutative 94.7%

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

       2. associate-*l/ 94.6%

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

       3. associate-*r/ 94.6%

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

       4. clear-num 94.6%

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

       5. un-div-inv 94.6%

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

       6. div-inv 94.8%

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

       7. metadata-eval 94.8%

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

   13. Applied egg-rr 94.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+82}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+38}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+25}:\\ \;\;\;\;1 + \frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e+25)
   (+ 1.0 (/ (- (* -0.3333333333333333 (* y (sqrt x))) 0.1111111111111111) x))
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e+25) {
		tmp = 1.0 + (((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.15d+25) then
        tmp = 1.0d0 + ((((-0.3333333333333333d0) * (y * sqrt(x))) - 0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.15e+25) {
		tmp = 1.0 + (((-0.3333333333333333 * (y * Math.sqrt(x))) - 0.1111111111111111) / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.15e+25:
		tmp = 1.0 + (((-0.3333333333333333 * (y * math.sqrt(x))) - 0.1111111111111111) / x)
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e+25)
		tmp = Float64(1.0 + Float64(Float64(Float64(-0.3333333333333333 * Float64(y * sqrt(x))) - 0.1111111111111111) / x));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.15e+25)
		tmp = 1.0 + (((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x);
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e+25], N[(1.0 + N[(N[(N[(-0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999e25

   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 x around 0 99.4%

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

   if 1.1499999999999999e25 < x

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-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* 99.8%

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

      15. distribute-neg-frac 99.8%

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

      16. metadata-eval 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.7%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*l* 99.8%

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

   7. Simplified 99.8%

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

      1. add-log-exp 59.7%

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

      2. *-un-lft-identity 59.7%

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

      3. log-prod 59.7%

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

      4. metadata-eval 59.7%

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

      5. add-log-exp 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.7%

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

      8. sqrt-div 99.8%

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

      9. metadata-eval 99.8%

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

      10. un-div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. +-lft-identity 99.8%

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

   11. Simplified 99.8%

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

       1. clear-num 99.7%

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

       2. un-div-inv 99.7%

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

   13. Applied egg-rr 99.7%

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

       1. associate-/r/ 99.8%

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

   15. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+25}:\\ \;\;\;\;1 + \frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 8: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1 - {\left(x \cdot 9\right)}^{-1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 - math.pow((x * 9.0), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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* 99.7%

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

   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* 99.7%

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

   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 63.1%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 30.1%

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

   3. frac-times 30.1%

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

   4. metadata-eval 30.1%

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

   5. metadata-eval 30.1%

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

   6. frac-times 30.1%

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

   7. clear-num 30.1%

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

   8. clear-num 30.1%

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

   9. frac-times 30.1%

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

   10. div-inv 30.1%

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

   11. metadata-eval 30.1%

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

   12. div-inv 30.1%

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

   13. metadata-eval 30.1%

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

   14. frac-times 30.1%

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

   15. sqrt-unprod 29.0%

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

   16. add-sqr-sqrt 29.0%

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

   17. metadata-eval 29.0%

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

   18. div-inv 29.0%

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

   19. frac-2neg 29.0%

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

   20. metadata-eval 29.0%

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

   21. clear-num 29.0%

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

   22. distribute-frac-neg2 29.0%

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

7. Applied egg-rr 29.0%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 49.0%

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

   3. frac-times 49.0%

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

   4. metadata-eval 49.0%

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

   5. metadata-eval 49.0%

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

   6. frac-times 49.0%

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

   7. clear-num 49.0%

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

   8. clear-num 49.0%

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

   9. frac-times 49.0%

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

   10. div-inv 49.0%

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

   11. metadata-eval 49.0%

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

   12. div-inv 49.0%

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

   13. metadata-eval 49.0%

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

   14. frac-times 49.0%

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

   15. sqrt-unprod 63.0%

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

   16. add-sqr-sqrt 63.1%

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

   17. inv-pow 63.1%

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

9. Applied egg-rr 63.1%

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

10. Final simplification 63.1%

    \[\leadsto 1 - {\left(x \cdot 9\right)}^{-1} \]
11. Add Preprocessing

Alternative 9: 62.4% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. 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* 99.7%

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

   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* 99.7%

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

   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 63.1%

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

6. Final simplification 63.1%

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

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024130 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))