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

Percentage Accurate: 99.7% → 99.7%

Time: 10.4s

Alternatives: 14

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-commutative 99.8%

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

   2. metadata-eval 99.8%

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

   3. sqrt-prod 99.8%

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

   4. pow1/2 99.8%

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

3. Applied egg-rr 99.8%

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

   1. unpow1/2 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+97}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{\sqrt{x}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e+104)
   (* -0.3333333333333333 (* y (pow x -0.5)))
   (if (<= y 2.5e+97)
     (- 1.0 (pow (* x 9.0) -1.0))
     (/ 1.0 (* -3.0 (/ (sqrt x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+104) {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	} else if (y <= 2.5e+97) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 / (-3.0 * (sqrt(x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.7d+104)) then
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    else if (y <= 2.5d+97) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 / ((-3.0d0) * (sqrt(x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+104) {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	} else if (y <= 2.5e+97) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 / (-3.0 * (Math.sqrt(x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.7e+104:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	elif y <= 2.5e+97:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 / (-3.0 * (math.sqrt(x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.7e+104)
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	elseif (y <= 2.5e+97)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 / Float64(-3.0 * Float64(sqrt(x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e+104)
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	elseif (y <= 2.5e+97)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 / (-3.0 * (sqrt(x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.7e+104], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+97], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-3.0 * N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999998e104

   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. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 94.2%

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

      1. expm1-log1p-u 93.0%

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

      2. expm1-udef 48.5%

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

      3. inv-pow 48.5%

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

      4. sqrt-pow1 48.5%

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

      5. metadata-eval 48.5%

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

   8. Applied egg-rr 48.5%

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

      1. expm1-def 93.1%

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

      2. expm1-log1p 94.4%

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

   10. Simplified 94.4%

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

   if -1.6999999999999998e104 < y < 2.49999999999999999e97

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 95.7%

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

   5. Taylor expanded in x around 0 95.8%

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

      1. clear-num 95.8%

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

      2. inv-pow 95.8%

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

      3. div-inv 95.8%

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

      4. metadata-eval 95.8%

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

   7. Applied egg-rr 95.8%

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

   if 2.49999999999999999e97 < 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. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 95.7%

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

      1. expm1-log1p-u 94.3%

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

      2. expm1-udef 51.2%

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

      3. inv-pow 51.2%

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

      4. sqrt-pow1 51.2%

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

      5. metadata-eval 51.2%

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

   8. Applied egg-rr 51.2%

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

      1. expm1-def 94.3%

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

      2. expm1-log1p 95.7%

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

   10. Simplified 95.7%

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

       1. *-commutative 95.7%

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

       2. associate-*l* 95.6%

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

       3. metadata-eval 95.6%

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

       4. pow-flip 95.6%

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

       5. pow1/2 95.6%

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

       6. *-commutative 95.6%

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

       7. associate-/r/ 95.7%

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

       8. *-un-lft-identity 95.7%

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

       9. times-frac 95.9%

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

       10. metadata-eval 95.9%

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

   12. Applied egg-rr 95.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+97}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{\sqrt{x}}{y}}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. clear-num 99.7%

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

   2. un-div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-/l* 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 0.1111111111111111 x)) (/ y (sqrt (* x 9.0)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) - (y / sqrt((x * 9.0d0)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / Math.sqrt((x * 9.0)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) - (y / math.sqrt((x * 9.0)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) - Float64(y / sqrt(Float64(x * 9.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. *-commutative 99.8%

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

   2. metadata-eval 99.8%

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

   3. sqrt-prod 99.8%

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

   4. pow1/2 99.8%

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

3. Applied egg-rr 99.8%

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

   1. unpow1/2 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 6: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+98}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e+104)
   (* -0.3333333333333333 (* y (pow x -0.5)))
   (if (<= y 1.6e+98)
     (- 1.0 (pow (* x 9.0) -1.0))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+104) {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	} else if (y <= 1.6e+98) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = (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 (y <= (-1.7d+104)) then
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    else if (y <= 1.6d+98) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+104) {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	} else if (y <= 1.6e+98) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.7e+104:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	elif y <= 1.6e+98:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.7e+104)
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	elseif (y <= 1.6e+98)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e+104)
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	elseif (y <= 1.6e+98)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.7e+104], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+98], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999998e104

   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. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 94.2%

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

      1. expm1-log1p-u 93.0%

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

      2. expm1-udef 48.5%

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

      3. inv-pow 48.5%

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

      4. sqrt-pow1 48.5%

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

      5. metadata-eval 48.5%

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

   8. Applied egg-rr 48.5%

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

      1. expm1-def 93.1%

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

      2. expm1-log1p 94.4%

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

   10. Simplified 94.4%

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

   if -1.6999999999999998e104 < y < 1.6000000000000001e98

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 95.7%

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

   5. Taylor expanded in x around 0 95.8%

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

      1. clear-num 95.8%

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

      2. inv-pow 95.8%

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

      3. div-inv 95.8%

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

      4. metadata-eval 95.8%

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

   7. Applied egg-rr 95.8%

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

   if 1.6000000000000001e98 < 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. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 95.7%

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

      1. expm1-log1p-u 94.3%

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

      2. expm1-udef 51.2%

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

      3. inv-pow 51.2%

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

      4. sqrt-pow1 51.2%

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

      5. metadata-eval 51.2%

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

   8. Applied egg-rr 51.2%

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

      1. expm1-def 94.3%

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

      2. expm1-log1p 95.7%

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

   10. Simplified 95.7%

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

       1. *-commutative 95.7%

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

       2. associate-*r* 95.6%

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

       3. metadata-eval 95.6%

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

       4. pow-flip 95.6%

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

       5. pow1/2 95.6%

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

       6. div-inv 95.9%

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

       7. *-commutative 95.9%

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

   12. Applied egg-rr 95.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+98}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 7: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+105} \lor \neg \left(y \leq 5.7 \cdot 10^{+97}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.25e+105) (not (<= y 5.7e+97)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.25e+105) || !(y <= 5.7e+97)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.25d+105)) .or. (.not. (y <= 5.7d+97))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.25e+105) || !(y <= 5.7e+97)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.25e+105) or not (y <= 5.7e+97):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.25e+105) || !(y <= 5.7e+97))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.25e+105) || ~((y <= 5.7e+97)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.25e+105], N[Not[LessEqual[y, 5.7e+97]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25000000000000011e105 or 5.7000000000000002e97 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 95.0%

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

      1. expm1-log1p-u 43.3%

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

      2. expm1-udef 43.3%

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

      3. associate-*r* 43.3%

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

      4. *-commutative 43.3%

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

      5. sqrt-div 43.3%

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

      6. metadata-eval 43.3%

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

      7. un-div-inv 43.3%

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

   8. Applied egg-rr 43.3%

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

      1. expm1-def 43.3%

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

      2. expm1-log1p 94.9%

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

      3. associate-*r/ 95.0%

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

      4. associate-*l/ 95.0%

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

      5. *-commutative 95.0%

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

   10. Simplified 95.0%

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

   if -1.25000000000000011e105 < y < 5.7000000000000002e97

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 95.7%

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

      1. cancel-sign-sub-inv 95.7%

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

      2. metadata-eval 95.7%

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

      3. associate-*r/ 95.8%

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

      4. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+105} \lor \neg \left(y \leq 5.7 \cdot 10^{+97}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 8: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+97}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e+104)
   (/ -0.3333333333333333 (/ (sqrt x) y))
   (if (<= y 1.95e+97)
     (+ 1.0 (/ -0.1111111111111111 x))
     (* -0.3333333333333333 (/ y (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+104) {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	} else if (y <= 1.95e+97) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.6d+104)) then
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    else if (y <= 1.95d+97) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+104) {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	} else if (y <= 1.95e+97) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.6e+104:
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	elif y <= 1.95e+97:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e+104)
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	elseif (y <= 1.95e+97)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.6e+104)
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	elseif (y <= 1.95e+97)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = -0.3333333333333333 * (y / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e+104], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.95e+97], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6e104

   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. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 94.2%

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

      1. expm1-log1p-u 93.0%

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

      2. expm1-udef 48.5%

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

      3. inv-pow 48.5%

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

      4. sqrt-pow1 48.5%

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

      5. metadata-eval 48.5%

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

   8. Applied egg-rr 48.5%

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

      1. expm1-def 93.1%

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

      2. expm1-log1p 94.4%

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

   10. Simplified 94.4%

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

       1. *-commutative 94.4%

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

       2. associate-*r* 94.2%

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

       3. metadata-eval 94.2%

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

       4. pow-flip 94.2%

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

       5. pow1/2 94.2%

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

       6. div-inv 94.1%

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

       7. associate-/l* 94.2%

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

   12. Applied egg-rr 94.2%

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

   if -2.6e104 < y < 1.95e97

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 95.7%

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

      1. cancel-sign-sub-inv 95.7%

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

      2. metadata-eval 95.7%

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

      3. associate-*r/ 95.8%

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

      4. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   if 1.95e97 < 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. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 95.7%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. associate-*r* 0.0%

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

      4. *-commutative 0.0%

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

      5. sqrt-div 0.0%

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

      6. metadata-eval 0.0%

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

      7. un-div-inv 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 95.8%

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

      3. associate-*r/ 95.9%

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

      4. associate-*l/ 95.8%

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

      5. *-commutative 95.8%

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

   10. Simplified 95.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+97}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 9: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8e+104)
   (/ -0.3333333333333333 (/ (sqrt x) y))
   (if (<= y 1.7e+97)
     (+ 1.0 (/ -0.1111111111111111 x))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8e+104) {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	} else if (y <= 1.7e+97) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (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 (y <= (-1.8d+104)) then
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    else if (y <= 1.7d+97) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.8e+104) {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	} else if (y <= 1.7e+97) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8e+104:
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	elif y <= 1.7e+97:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8e+104)
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	elseif (y <= 1.7e+97)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.8e+104)
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	elseif (y <= 1.7e+97)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8e+104], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+97], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e104

   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. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 94.2%

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

      1. expm1-log1p-u 93.0%

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

      2. expm1-udef 48.5%

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

      3. inv-pow 48.5%

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

      4. sqrt-pow1 48.5%

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

      5. metadata-eval 48.5%

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

   8. Applied egg-rr 48.5%

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

      1. expm1-def 93.1%

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

      2. expm1-log1p 94.4%

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

   10. Simplified 94.4%

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

       1. *-commutative 94.4%

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

       2. associate-*r* 94.2%

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

       3. metadata-eval 94.2%

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

       4. pow-flip 94.2%

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

       5. pow1/2 94.2%

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

       6. div-inv 94.1%

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

       7. associate-/l* 94.2%

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

   12. Applied egg-rr 94.2%

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

   if -1.8e104 < y < 1.70000000000000005e97

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 95.7%

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

      1. cancel-sign-sub-inv 95.7%

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

      2. metadata-eval 95.7%

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

      3. associate-*r/ 95.8%

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

      4. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   if 1.70000000000000005e97 < 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. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 95.7%

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

      1. expm1-log1p-u 94.3%

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

      2. expm1-udef 51.2%

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

      3. inv-pow 51.2%

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

      4. sqrt-pow1 51.2%

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

      5. metadata-eval 51.2%

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

   8. Applied egg-rr 51.2%

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

      1. expm1-def 94.3%

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

      2. expm1-log1p 95.7%

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

   10. Simplified 95.7%

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

       1. *-commutative 95.7%

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

       2. associate-*r* 95.6%

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

       3. metadata-eval 95.6%

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

       4. pow-flip 95.6%

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

       5. pow1/2 95.6%

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

       6. div-inv 95.9%

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

       7. *-commutative 95.9%

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

   12. Applied egg-rr 95.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 10: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+104}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e+104)
   (* -0.3333333333333333 (* y (pow x -0.5)))
   (if (<= y 1.25e+99)
     (+ 1.0 (/ -0.1111111111111111 x))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+104) {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	} else if (y <= 1.25e+99) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (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 (y <= (-2.5d+104)) then
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    else if (y <= 1.25d+99) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+104) {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	} else if (y <= 1.25e+99) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e+104:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	elif y <= 1.25e+99:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e+104)
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	elseif (y <= 1.25e+99)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e+104)
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	elseif (y <= 1.25e+99)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e+104], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+99], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4999999999999998e104

   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. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 94.2%

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

      1. expm1-log1p-u 93.0%

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

      2. expm1-udef 48.5%

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

      3. inv-pow 48.5%

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

      4. sqrt-pow1 48.5%

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

      5. metadata-eval 48.5%

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

   8. Applied egg-rr 48.5%

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

      1. expm1-def 93.1%

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

      2. expm1-log1p 94.4%

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

   10. Simplified 94.4%

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

   if -2.4999999999999998e104 < y < 1.25000000000000002e99

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 95.7%

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

      1. cancel-sign-sub-inv 95.7%

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

      2. metadata-eval 95.7%

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

      3. associate-*r/ 95.8%

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

      4. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   if 1.25000000000000002e99 < 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. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 95.7%

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

      1. expm1-log1p-u 94.3%

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

      2. expm1-udef 51.2%

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

      3. inv-pow 51.2%

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

      4. sqrt-pow1 51.2%

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

      5. metadata-eval 51.2%

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

   8. Applied egg-rr 51.2%

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

      1. expm1-def 94.3%

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

      2. expm1-log1p 95.7%

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

   10. Simplified 95.7%

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

       1. *-commutative 95.7%

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

       2. associate-*r* 95.6%

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

       3. metadata-eval 95.6%

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

       4. pow-flip 95.6%

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

       5. pow1/2 95.6%

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

       6. div-inv 95.9%

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

       7. *-commutative 95.9%

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

   12. Applied egg-rr 95.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+104}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+99}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 11: 64.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+124}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{0.1111111111111111}{x \cdot \left(x \cdot 9\right)}}{\frac{0.1111111111111111}{x} + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7e+124)
   (+ 1.0 (/ -0.1111111111111111 x))
   (/
    (- -1.0 (/ 0.1111111111111111 (* x (* x 9.0))))
    (+ (/ 0.1111111111111111 x) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e+124) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (-1.0 - (0.1111111111111111 / (x * (x * 9.0)))) / ((0.1111111111111111 / x) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7d+124) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = ((-1.0d0) - (0.1111111111111111d0 / (x * (x * 9.0d0)))) / ((0.1111111111111111d0 / x) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7e+124) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (-1.0 - (0.1111111111111111 / (x * (x * 9.0)))) / ((0.1111111111111111 / x) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e+124:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (-1.0 - (0.1111111111111111 / (x * (x * 9.0)))) / ((0.1111111111111111 / x) + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e+124)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(-1.0 - Float64(0.1111111111111111 / Float64(x * Float64(x * 9.0)))) / Float64(Float64(0.1111111111111111 / x) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e+124)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (-1.0 - (0.1111111111111111 / (x * (x * 9.0)))) / ((0.1111111111111111 / x) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e+124], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 - N[(0.1111111111111111 / N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.0000000000000002e124

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 77.5%

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

      1. cancel-sign-sub-inv 77.5%

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

      2. metadata-eval 77.5%

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

      3. associate-*r/ 77.6%

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

      4. metadata-eval 77.6%

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

   6. Simplified 77.6%

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

   if 7.0000000000000002e124 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-/r* 99.6%

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

      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 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 4.1%

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

   6. Applied egg-rr 32.2%

      \[\leadsto \color{blue}{\frac{-\left(1 + {x}^{-2} \cdot 0.012345679012345678\right)}{-\left(1 + \frac{-0.1111111111111111}{x}\right)}} \]
   7. Step-by-step derivation

      1. distribute-neg-in 32.2%

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

      2. metadata-eval 32.2%

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

      3. unsub-neg 32.2%

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

      4. distribute-neg-in 32.2%

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

      5. metadata-eval 32.2%

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

      6. distribute-neg-frac 32.2%

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

      7. metadata-eval 32.2%

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

   8. Simplified 32.2%

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

      1. *-commutative 32.2%

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

      2. metadata-eval 32.2%

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

      3. metadata-eval 32.2%

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

      4. pow-sqr 32.2%

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

      5. inv-pow 32.2%

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

      6. inv-pow 32.2%

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

      7. swap-sqr 32.2%

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

      8. div-inv 32.2%

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

      9. div-inv 32.2%

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

      10. clear-num 32.2%

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

      11. frac-times 32.2%

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

      12. metadata-eval 32.2%

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

      13. div-inv 32.2%

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

      14. metadata-eval 32.2%

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

   10. Applied egg-rr 32.2%

       \[\leadsto \frac{-1 - \color{blue}{\frac{0.1111111111111111}{\left(x \cdot 9\right) \cdot x}}}{-1 + \frac{0.1111111111111111}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+124}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{0.1111111111111111}{x \cdot \left(x \cdot 9\right)}}{\frac{0.1111111111111111}{x} + -1}\\ \end{array} \]

Alternative 12: 61.3% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2300:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2300.0) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2300.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2300.0d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2300.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2300.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2300.0)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2300.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2300.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2300

   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. Taylor expanded in x around 0 65.4%

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

   if 2300 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 68.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2300:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 62.3% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in y around 0 67.1%

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

   1. cancel-sign-sub-inv 67.1%

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

   2. metadata-eval 67.1%

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

   3. associate-*r/ 67.2%

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

   4. metadata-eval 67.2%

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

6. Simplified 67.2%

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

7. Final simplification 67.2%

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

Alternative 14: 31.7% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around inf 36.8%

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

5. Final simplification 36.8%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023318 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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