Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Percentage Accurate: 99.4% → 99.4%

Time: 8.4s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in y around 0 99.4%

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

   1. div-inv 99.4%

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

   2. clear-num 99.4%

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

   3. div-inv 99.4%

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

   4. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-32}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x))))
   (if (<= x 1.5e-78)
     t_0
     (if (<= x 6.8e-32)
       (* y (* (sqrt x) 3.0))
       (if (<= x 1.25e-12) t_0 (* (sqrt x) (- (* 3.0 y) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (x <= 1.5e-78) {
		tmp = t_0;
	} else if (x <= 6.8e-32) {
		tmp = y * (sqrt(x) * 3.0);
	} else if (x <= 1.25e-12) {
		tmp = t_0;
	} else {
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    if (x <= 1.5d-78) then
        tmp = t_0
    else if (x <= 6.8d-32) then
        tmp = y * (sqrt(x) * 3.0d0)
    else if (x <= 1.25d-12) then
        tmp = t_0
    else
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((0.1111111111111111 / x));
	double tmp;
	if (x <= 1.5e-78) {
		tmp = t_0;
	} else if (x <= 6.8e-32) {
		tmp = y * (Math.sqrt(x) * 3.0);
	} else if (x <= 1.25e-12) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if x <= 1.5e-78:
		tmp = t_0
	elif x <= 6.8e-32:
		tmp = y * (math.sqrt(x) * 3.0)
	elif x <= 1.25e-12:
		tmp = t_0
	else:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (x <= 1.5e-78)
		tmp = t_0;
	elseif (x <= 6.8e-32)
		tmp = Float64(y * Float64(sqrt(x) * 3.0));
	elseif (x <= 1.25e-12)
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) - 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (x <= 1.5e-78)
		tmp = t_0;
	elseif (x <= 6.8e-32)
		tmp = y * (sqrt(x) * 3.0);
	elseif (x <= 1.25e-12)
		tmp = t_0;
	else
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.5e-78], t$95$0, If[LessEqual[x, 6.8e-32], N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-12], t$95$0, N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.49999999999999994e-78 or 6.79999999999999956e-32 < x < 1.24999999999999992e-12

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-rgt-in 99.3%

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

      4. remove-double-neg 99.3%

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

      5. distribute-lft-neg-in 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. mul-1-neg 99.3%

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

      8. metadata-eval 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-/r* 99.2%

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

      11. distribute-neg-frac 99.2%

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

      12. *-commutative 99.2%

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

      13. associate-/r/ 99.3%

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

      14. associate-/l/ 99.3%

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

      15. associate-/r/ 99.2%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 78.5%

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

      1. *-commutative 78.5%

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

      2. clear-num 78.4%

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

      3. un-div-inv 78.4%

         \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 3}{\frac{x}{0.1111111111111111}}} \]

      4. div-inv 78.6%

         \[\leadsto \frac{\sqrt{x} \cdot 3}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      5. metadata-eval 78.6%

         \[\leadsto \frac{\sqrt{x} \cdot 3}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 78.6%

      \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 3}{x \cdot 9}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 78.6%

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

      2. add-sqr-sqrt 78.5%

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

      3. metadata-eval 78.5%

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

      4. swap-sqr 78.5%

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

      5. times-frac 78.6%

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

      6. metadata-eval 78.6%

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

      7. metadata-eval 78.6%

         \[\leadsto \frac{\sqrt{1}}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      8. sqrt-prod 78.7%

         \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{x \cdot 9}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      9. metadata-eval 78.7%

         \[\leadsto \frac{\sqrt{1}}{\sqrt{x \cdot \color{blue}{\frac{1}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      10. div-inv 78.6%

          \[\leadsto \frac{\sqrt{1}}{\sqrt{\color{blue}{\frac{x}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      11. sqrt-div 78.8%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{x}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      12. clear-num 78.9%

          \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      13. sqrt-div 78.7%

          \[\leadsto \color{blue}{\frac{\sqrt{0.1111111111111111}}{\sqrt{x}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      14. metadata-eval 78.7%

          \[\leadsto \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      15. *-commutative 78.7%

          \[\leadsto \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{\color{blue}{3 \cdot \sqrt{x}}}{\sqrt{x} \cdot 3} \]

      16. *-commutative 78.7%

          \[\leadsto \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{3 \cdot \sqrt{x}}{\color{blue}{3 \cdot \sqrt{x}}} \]

   8. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{3 \cdot \sqrt{x}}{3 \cdot \sqrt{x}}} \]
   9. Step-by-step derivation

      1. *-inverses 78.7%

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

      2. *-rgt-identity 78.7%

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

   10. Simplified 78.7%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
   11. Step-by-step derivation

       1. metadata-eval 78.7%

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

       2. sqrt-div 78.9%

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

       3. pow1/2 78.9%

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

   12. Applied egg-rr 78.9%

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

       1. unpow1/2 78.9%

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

   14. Simplified 78.9%

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

   if 1.49999999999999994e-78 < x < 6.79999999999999956e-32

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-rgt-in 99.3%

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

      4. remove-double-neg 99.3%

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

      5. distribute-lft-neg-in 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. mul-1-neg 99.3%

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

      8. metadata-eval 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-/r* 99.2%

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

      11. distribute-neg-frac 99.2%

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

      12. *-commutative 99.2%

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

      13. associate-/r/ 99.0%

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

      14. associate-/l/ 99.0%

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

      15. associate-/r/ 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around inf 74.7%

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

   if 1.24999999999999992e-12 < x

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 98.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-32}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]

Alternative 3: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{x} \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2e+41)
   (* (sqrt x) (* 3.0 y))
   (if (<= y 4.5e-20)
     (* (sqrt x) (- (* 0.3333333333333333 (/ 1.0 x)) 3.0))
     (* (* (sqrt x) 3.0) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+41) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= 4.5e-20) {
		tmp = sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	} else {
		tmp = (sqrt(x) * 3.0) * (y + -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 <= (-7.2d+41)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= 4.5d-20) then
        tmp = sqrt(x) * ((0.3333333333333333d0 * (1.0d0 / x)) - 3.0d0)
    else
        tmp = (sqrt(x) * 3.0d0) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+41) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= 4.5e-20) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	} else {
		tmp = (Math.sqrt(x) * 3.0) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e+41:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= 4.5e-20:
		tmp = math.sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0)
	else:
		tmp = (math.sqrt(x) * 3.0) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e+41)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= 4.5e-20)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 3.0));
	else
		tmp = Float64(Float64(sqrt(x) * 3.0) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.2e+41)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= 4.5e-20)
		tmp = sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	else
		tmp = (sqrt(x) * 3.0) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e+41], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-20], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.20000000000000051e41

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 73.2%

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

      1. associate-*r* 73.2%

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

      2. *-commutative 73.2%

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

   6. Simplified 73.2%

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

   if -7.20000000000000051e41 < y < 4.5000000000000001e-20

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 97.1%

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

   if 4.5000000000000001e-20 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.6%

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

      11. distribute-neg-frac 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r/ 99.6%

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

      14. associate-/l/ 99.6%

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

      15. associate-/r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 84.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{x} \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 4: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.42e+41)
   (* (sqrt x) (* 3.0 y))
   (if (<= y 4.5e-20)
     (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
     (* (* (sqrt x) 3.0) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.42e+41) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= 4.5e-20) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = (sqrt(x) * 3.0) * (y + -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 <= (-1.42d+41)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= 4.5d-20) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = (sqrt(x) * 3.0d0) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.42e+41) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= 4.5e-20) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = (Math.sqrt(x) * 3.0) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.42e+41:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= 4.5e-20:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = (math.sqrt(x) * 3.0) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.42e+41)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= 4.5e-20)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(Float64(sqrt(x) * 3.0) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.42e+41)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= 4.5e-20)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = (sqrt(x) * 3.0) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.42e+41], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-20], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.42000000000000007e41

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 73.2%

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

      1. associate-*r* 73.2%

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

      2. *-commutative 73.2%

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

   6. Simplified 73.2%

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

   if -1.42000000000000007e41 < y < 4.5000000000000001e-20

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 97.1%

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

      1. sub-neg 97.1%

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

      2. associate-*r/ 97.1%

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

      3. metadata-eval 97.1%

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

      4. metadata-eval 97.1%

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

   6. Simplified 97.1%

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

   if 4.5000000000000001e-20 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.6%

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

      11. distribute-neg-frac 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r/ 99.6%

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

      14. associate-/l/ 99.6%

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

      15. associate-/r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 84.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot \left(\left(3 \cdot y + \frac{0.3333333333333333}{x}\right) - 3\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sqrt x) (- (+ (* 3.0 y) (/ 0.3333333333333333 x)) 3.0)))

C

double code(double x, double y) {
	return sqrt(x) * (((3.0 * y) + (0.3333333333333333 / x)) - 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * (((3.0d0 * y) + (0.3333333333333333d0 / x)) - 3.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x) * (((3.0 * y) + (0.3333333333333333 / x)) - 3.0);
}

Python

def code(x, y):
	return math.sqrt(x) * (((3.0 * y) + (0.3333333333333333 / x)) - 3.0)

Julia

function code(x, y)
	return Float64(sqrt(x) * Float64(Float64(Float64(3.0 * y) + Float64(0.3333333333333333 / x)) - 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * (((3.0 * y) + (0.3333333333333333 / x)) - 3.0);
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[(3.0 * y), $MachinePrecision] + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in y around 0 99.4%

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

5. Taylor expanded in x around 0 99.4%

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

6. Final simplification 99.4%

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

Alternative 6: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -33000000000 \lor \neg \left(y \leq 6.7 \cdot 10^{+49}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -33000000000.0) (not (<= y 6.7e+49)))
   (* 3.0 (* (sqrt x) y))
   (sqrt (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -33000000000.0) || !(y <= 6.7e+49)) {
		tmp = 3.0 * (sqrt(x) * y);
	} else {
		tmp = sqrt((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 <= (-33000000000.0d0)) .or. (.not. (y <= 6.7d+49))) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else
        tmp = sqrt((0.1111111111111111d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -33000000000.0) || !(y <= 6.7e+49)) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt((0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -33000000000.0) or not (y <= 6.7e+49):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt((0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -33000000000.0) || !(y <= 6.7e+49))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = sqrt(Float64(0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -33000000000.0) || ~((y <= 6.7e+49)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt((0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -33000000000.0], N[Not[LessEqual[y, 6.7e+49]], $MachinePrecision]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e10 or 6.6999999999999999e49 < y

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 79.6%

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

   if -3.3e10 < y < 6.6999999999999999e49

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-rgt-in 99.3%

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

      4. remove-double-neg 99.3%

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

      5. distribute-lft-neg-in 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. mul-1-neg 99.3%

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

      8. metadata-eval 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-/r* 99.3%

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

      11. distribute-neg-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/r/ 99.3%

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

      14. associate-/l/ 99.3%

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

      15. associate-/r/ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 48.5%

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

      1. *-commutative 48.5%

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

      2. clear-num 48.5%

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

      3. un-div-inv 48.5%

         \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 3}{\frac{x}{0.1111111111111111}}} \]

      4. div-inv 48.6%

         \[\leadsto \frac{\sqrt{x} \cdot 3}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      5. metadata-eval 48.6%

         \[\leadsto \frac{\sqrt{x} \cdot 3}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 3}{x \cdot 9}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 48.6%

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

      2. add-sqr-sqrt 48.5%

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

      3. metadata-eval 48.5%

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

      4. swap-sqr 48.5%

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

      5. times-frac 48.6%

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

      6. metadata-eval 48.6%

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

      7. metadata-eval 48.6%

         \[\leadsto \frac{\sqrt{1}}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      8. sqrt-prod 48.7%

         \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{x \cdot 9}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      9. metadata-eval 48.7%

         \[\leadsto \frac{\sqrt{1}}{\sqrt{x \cdot \color{blue}{\frac{1}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      10. div-inv 48.7%

          \[\leadsto \frac{\sqrt{1}}{\sqrt{\color{blue}{\frac{x}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      11. sqrt-div 48.7%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{x}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      12. clear-num 48.7%

          \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      13. sqrt-div 48.6%

          \[\leadsto \color{blue}{\frac{\sqrt{0.1111111111111111}}{\sqrt{x}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      14. metadata-eval 48.6%

          \[\leadsto \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      15. *-commutative 48.6%

          \[\leadsto \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{\color{blue}{3 \cdot \sqrt{x}}}{\sqrt{x} \cdot 3} \]

      16. *-commutative 48.6%

          \[\leadsto \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{3 \cdot \sqrt{x}}{\color{blue}{3 \cdot \sqrt{x}}} \]

   8. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{3 \cdot \sqrt{x}}{3 \cdot \sqrt{x}}} \]
   9. Step-by-step derivation

      1. *-inverses 48.6%

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

      2. *-rgt-identity 48.6%

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

   10. Simplified 48.6%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
   11. Step-by-step derivation

       1. metadata-eval 48.6%

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

       2. sqrt-div 48.7%

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

       3. pow1/2 48.7%

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

   12. Applied egg-rr 48.7%

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

       1. unpow1/2 48.7%

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

   14. Simplified 48.7%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -33000000000 \lor \neg \left(y \leq 6.7 \cdot 10^{+49}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \]

Alternative 7: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+20} \lor \neg \left(y \leq 1.35 \cdot 10^{+51}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.2e+20) (not (<= y 1.35e+51)))
   (* (sqrt x) (* 3.0 y))
   (sqrt (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+20) || !(y <= 1.35e+51)) {
		tmp = sqrt(x) * (3.0 * y);
	} else {
		tmp = sqrt((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 <= (-4.2d+20)) .or. (.not. (y <= 1.35d+51))) then
        tmp = sqrt(x) * (3.0d0 * y)
    else
        tmp = sqrt((0.1111111111111111d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+20) || !(y <= 1.35e+51)) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else {
		tmp = Math.sqrt((0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.2e+20) or not (y <= 1.35e+51):
		tmp = math.sqrt(x) * (3.0 * y)
	else:
		tmp = math.sqrt((0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.2e+20) || !(y <= 1.35e+51))
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	else
		tmp = sqrt(Float64(0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.2e+20) || ~((y <= 1.35e+51)))
		tmp = sqrt(x) * (3.0 * y);
	else
		tmp = sqrt((0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.2e+20], N[Not[LessEqual[y, 1.35e+51]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e20 or 1.34999999999999996e51 < y

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 79.6%

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

      1. associate-*r* 79.7%

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

      2. *-commutative 79.7%

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

   6. Simplified 79.7%

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

   if -4.2e20 < y < 1.34999999999999996e51

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-rgt-in 99.3%

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

      4. remove-double-neg 99.3%

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

      5. distribute-lft-neg-in 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. mul-1-neg 99.3%

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

      8. metadata-eval 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-/r* 99.3%

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

      11. distribute-neg-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/r/ 99.3%

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

      14. associate-/l/ 99.3%

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

      15. associate-/r/ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 48.5%

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

      1. *-commutative 48.5%

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

      2. clear-num 48.5%

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

      3. un-div-inv 48.5%

         \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 3}{\frac{x}{0.1111111111111111}}} \]

      4. div-inv 48.6%

         \[\leadsto \frac{\sqrt{x} \cdot 3}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      5. metadata-eval 48.6%

         \[\leadsto \frac{\sqrt{x} \cdot 3}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 3}{x \cdot 9}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 48.6%

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

      2. add-sqr-sqrt 48.5%

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

      3. metadata-eval 48.5%

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

      4. swap-sqr 48.5%

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

      5. times-frac 48.6%

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

      6. metadata-eval 48.6%

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

      7. metadata-eval 48.6%

         \[\leadsto \frac{\sqrt{1}}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      8. sqrt-prod 48.7%

         \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{x \cdot 9}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      9. metadata-eval 48.7%

         \[\leadsto \frac{\sqrt{1}}{\sqrt{x \cdot \color{blue}{\frac{1}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      10. div-inv 48.7%

          \[\leadsto \frac{\sqrt{1}}{\sqrt{\color{blue}{\frac{x}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      11. sqrt-div 48.7%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{x}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      12. clear-num 48.7%

          \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      13. sqrt-div 48.6%

          \[\leadsto \color{blue}{\frac{\sqrt{0.1111111111111111}}{\sqrt{x}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      14. metadata-eval 48.6%

          \[\leadsto \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

      15. *-commutative 48.6%

          \[\leadsto \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{\color{blue}{3 \cdot \sqrt{x}}}{\sqrt{x} \cdot 3} \]

      16. *-commutative 48.6%

          \[\leadsto \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{3 \cdot \sqrt{x}}{\color{blue}{3 \cdot \sqrt{x}}} \]

   8. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{3 \cdot \sqrt{x}}{3 \cdot \sqrt{x}}} \]
   9. Step-by-step derivation

      1. *-inverses 48.6%

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

      2. *-rgt-identity 48.6%

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

   10. Simplified 48.6%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
   11. Step-by-step derivation

       1. metadata-eval 48.6%

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

       2. sqrt-div 48.7%

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

       3. pow1/2 48.7%

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

   12. Applied egg-rr 48.7%

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

       1. unpow1/2 48.7%

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

   14. Simplified 48.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+20} \lor \neg \left(y \leq 1.35 \cdot 10^{+51}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \]

Alternative 8: 38.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.1111111111111111}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (/ 0.1111111111111111 x)))

C

double code(double x, double y) {
	return sqrt((0.1111111111111111 / x));
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.sqrt((0.1111111111111111 / x))

Julia

function code(x, y)
	return sqrt(Float64(0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((0.1111111111111111 / x));
end

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. distribute-rgt-in 99.4%

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

   4. remove-double-neg 99.4%

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

   5. distribute-lft-neg-in 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. mul-1-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. *-commutative 99.4%

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

   10. associate-/r* 99.4%

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

   11. distribute-neg-frac 99.4%

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

   12. *-commutative 99.4%

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

   13. associate-/r/ 99.4%

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

   14. associate-/l/ 99.4%

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

   15. associate-/r/ 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 35.5%

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

   1. *-commutative 35.5%

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

   2. clear-num 35.5%

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

   3. un-div-inv 35.5%

      \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 3}{\frac{x}{0.1111111111111111}}} \]

   4. div-inv 35.6%

      \[\leadsto \frac{\sqrt{x} \cdot 3}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

   5. metadata-eval 35.6%

      \[\leadsto \frac{\sqrt{x} \cdot 3}{x \cdot \color{blue}{9}} \]

6. Applied egg-rr 35.6%

   \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 3}{x \cdot 9}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 35.6%

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

   2. add-sqr-sqrt 35.5%

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

   3. metadata-eval 35.5%

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

   4. swap-sqr 35.6%

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

   5. times-frac 35.6%

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

   6. metadata-eval 35.6%

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

   7. metadata-eval 35.6%

      \[\leadsto \frac{\sqrt{1}}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

   8. sqrt-prod 35.6%

      \[\leadsto \frac{\sqrt{1}}{\color{blue}{\sqrt{x \cdot 9}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

   9. metadata-eval 35.6%

      \[\leadsto \frac{\sqrt{1}}{\sqrt{x \cdot \color{blue}{\frac{1}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

   10. div-inv 35.6%

       \[\leadsto \frac{\sqrt{1}}{\sqrt{\color{blue}{\frac{x}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

   11. sqrt-div 35.7%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{x}{0.1111111111111111}}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

   12. clear-num 35.7%

       \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

   13. sqrt-div 35.6%

       \[\leadsto \color{blue}{\frac{\sqrt{0.1111111111111111}}{\sqrt{x}}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

   14. metadata-eval 35.6%

       \[\leadsto \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}} \cdot \frac{\sqrt{x} \cdot 3}{\sqrt{x} \cdot 3} \]

   15. *-commutative 35.6%

       \[\leadsto \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{\color{blue}{3 \cdot \sqrt{x}}}{\sqrt{x} \cdot 3} \]

   16. *-commutative 35.6%

       \[\leadsto \frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{3 \cdot \sqrt{x}}{\color{blue}{3 \cdot \sqrt{x}}} \]

8. Applied egg-rr 35.6%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}} \cdot \frac{3 \cdot \sqrt{x}}{3 \cdot \sqrt{x}}} \]
9. Step-by-step derivation

   1. *-inverses 35.6%

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

   2. *-rgt-identity 35.6%

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

10. Simplified 35.6%

    \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
11. Step-by-step derivation

    1. metadata-eval 35.6%

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

    2. sqrt-div 35.7%

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

    3. pow1/2 35.7%

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

12. Applied egg-rr 35.7%

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

    1. unpow1/2 35.7%

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

14. Simplified 35.7%

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

15. Final simplification 35.7%

    \[\leadsto \sqrt{\frac{0.1111111111111111}{x}} \]

Developer target: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023195 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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