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

Percentage Accurate: 99.4% → 99.4%

Time: 13.2s

Alternatives: 11

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*l* 99.6%

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

   3. associate--l+ 99.6%

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

   4. distribute-lft-in 99.6%

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

   5. fma-def 99.5%

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

   6. sub-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. distribute-lft-in 99.5%

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

   9. metadata-eval 99.5%

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

   10. metadata-eval 99.5%

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

   11. *-commutative 99.5%

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

   12. associate-/r* 99.5%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 59.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ t_1 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_2 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -8 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.35 \cdot 10^{-225}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-297}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-252}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2100000000 \lor \neg \left(y \leq 1.7 \cdot 10^{+59}\right) \land y \leq 2.35 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x)))
        (t_1 (* 3.0 (* (sqrt x) y)))
        (t_2 (* (sqrt x) -3.0)))
   (if (<= y -8e-6)
     t_1
     (if (<= y -4.35e-225)
       t_2
       (if (<= y 3.7e-297)
         t_0
         (if (<= y 5e-252)
           t_2
           (if (or (<= y 2100000000.0)
                   (and (not (<= y 1.7e+59)) (<= y 2.35e+113)))
             t_0
             t_1)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double t_1 = 3.0 * (sqrt(x) * y);
	double t_2 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -8e-6) {
		tmp = t_1;
	} else if (y <= -4.35e-225) {
		tmp = t_2;
	} else if (y <= 3.7e-297) {
		tmp = t_0;
	} else if (y <= 5e-252) {
		tmp = t_2;
	} else if ((y <= 2100000000.0) || (!(y <= 1.7e+59) && (y <= 2.35e+113))) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    t_1 = 3.0d0 * (sqrt(x) * y)
    t_2 = sqrt(x) * (-3.0d0)
    if (y <= (-8d-6)) then
        tmp = t_1
    else if (y <= (-4.35d-225)) then
        tmp = t_2
    else if (y <= 3.7d-297) then
        tmp = t_0
    else if (y <= 5d-252) then
        tmp = t_2
    else if ((y <= 2100000000.0d0) .or. (.not. (y <= 1.7d+59)) .and. (y <= 2.35d+113)) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((0.1111111111111111 / x));
	double t_1 = 3.0 * (Math.sqrt(x) * y);
	double t_2 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -8e-6) {
		tmp = t_1;
	} else if (y <= -4.35e-225) {
		tmp = t_2;
	} else if (y <= 3.7e-297) {
		tmp = t_0;
	} else if (y <= 5e-252) {
		tmp = t_2;
	} else if ((y <= 2100000000.0) || (!(y <= 1.7e+59) && (y <= 2.35e+113))) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	t_1 = 3.0 * (math.sqrt(x) * y)
	t_2 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -8e-6:
		tmp = t_1
	elif y <= -4.35e-225:
		tmp = t_2
	elif y <= 3.7e-297:
		tmp = t_0
	elif y <= 5e-252:
		tmp = t_2
	elif (y <= 2100000000.0) or (not (y <= 1.7e+59) and (y <= 2.35e+113)):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	t_1 = Float64(3.0 * Float64(sqrt(x) * y))
	t_2 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -8e-6)
		tmp = t_1;
	elseif (y <= -4.35e-225)
		tmp = t_2;
	elseif (y <= 3.7e-297)
		tmp = t_0;
	elseif (y <= 5e-252)
		tmp = t_2;
	elseif ((y <= 2100000000.0) || (!(y <= 1.7e+59) && (y <= 2.35e+113)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	t_1 = 3.0 * (sqrt(x) * y);
	t_2 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -8e-6)
		tmp = t_1;
	elseif (y <= -4.35e-225)
		tmp = t_2;
	elseif (y <= 3.7e-297)
		tmp = t_0;
	elseif (y <= 5e-252)
		tmp = t_2;
	elseif ((y <= 2100000000.0) || (~((y <= 1.7e+59)) && (y <= 2.35e+113)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -8e-6], t$95$1, If[LessEqual[y, -4.35e-225], t$95$2, If[LessEqual[y, 3.7e-297], t$95$0, If[LessEqual[y, 5e-252], t$95$2, If[Or[LessEqual[y, 2100000000.0], And[N[Not[LessEqual[y, 1.7e+59]], $MachinePrecision], LessEqual[y, 2.35e+113]]], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.99999999999999964e-6 or 2.1e9 < y < 1.70000000000000003e59 or 2.3499999999999999e113 < y

   1. Initial program 99.5%

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

      1. associate-*l* 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 79.3%

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

      1. *-commutative 79.3%

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

   6. Simplified 79.3%

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

   if -7.99999999999999964e-6 < y < -4.3500000000000003e-225 or 3.7e-297 < y < 5.00000000000000008e-252

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. *-commutative 97.3%

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

      2. sub-neg 97.3%

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

      3. metadata-eval 97.3%

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

      4. associate-*r/ 97.5%

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

      5. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in x around inf 61.0%

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

   if -4.3500000000000003e-225 < y < 3.7e-297 or 5.00000000000000008e-252 < y < 2.1e9 or 1.70000000000000003e59 < y < 2.3499999999999999e113

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 58.1%

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

      1. expm1-log1p-u 54.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      2. expm1-udef 54.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)} - 1} \]

      3. associate-*r* 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}}\right)} - 1 \]

      4. *-commutative 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      5. metadata-eval 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      6. sqrt-prod 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      7. add-sqr-sqrt 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 9} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      8. metadata-eval 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(-3 \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      9. swap-sqr 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      10. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\sqrt{x} \cdot -3} \cdot \sqrt{\sqrt{x} \cdot -3}\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      11. add-sqr-sqrt 1.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot -3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   6. Applied egg-rr 1.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 1.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-log1p 2.2%

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

      3. *-commutative 2.2%

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

      4. associate-*r* 2.2%

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

      5. associate-*l/ 2.2%

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

      6. *-commutative 2.2%

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

      7. rem-square-sqrt 2.2%

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

      8. times-frac 2.2%

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

      9. *-inverses 2.2%

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

      10. *-lft-identity 2.2%

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

      11. metadata-eval 2.2%

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

      12. metadata-eval 2.2%

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

      13. *-inverses 2.2%

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

      14. associate-/l* 2.2%

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

      15. associate-*l/ 2.2%

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

      16. associate-*r/ 2.2%

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

      17. associate-*r* 2.2%

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

      18. *-commutative 2.2%

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

      19. associate-*r* 2.2%

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

      20. associate-*r* 2.2%

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

      21. *-commutative 2.2%

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

   8. Simplified 2.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 58.4%

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

      3. frac-times 58.4%

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

      4. metadata-eval 58.4%

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

      5. add-sqr-sqrt 58.5%

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

      6. pow1/2 58.5%

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

   10. Applied egg-rr 58.5%

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

       1. unpow1/2 58.5%

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

   12. Simplified 58.5%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-6}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -4.35 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 2100000000 \lor \neg \left(y \leq 1.7 \cdot 10^{+59}\right) \land y \leq 2.35 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 3: 59.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ t_1 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -8 \cdot 10^{-6}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-297}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 56000000000 \lor \neg \left(y \leq 5.9 \cdot 10^{+56}\right) \land y \leq 2.35 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x))) (t_1 (* (sqrt x) -3.0)))
   (if (<= y -8e-6)
     (* 3.0 (* (sqrt x) y))
     (if (<= y -1.26e-225)
       t_1
       (if (<= y 6.5e-297)
         t_0
         (if (<= y 9.5e-252)
           t_1
           (if (or (<= y 56000000000.0)
                   (and (not (<= y 5.9e+56)) (<= y 2.35e+113)))
             t_0
             (* (sqrt x) (* 3.0 y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double t_1 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -8e-6) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= -1.26e-225) {
		tmp = t_1;
	} else if (y <= 6.5e-297) {
		tmp = t_0;
	} else if (y <= 9.5e-252) {
		tmp = t_1;
	} else if ((y <= 56000000000.0) || (!(y <= 5.9e+56) && (y <= 2.35e+113))) {
		tmp = t_0;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    t_1 = sqrt(x) * (-3.0d0)
    if (y <= (-8d-6)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= (-1.26d-225)) then
        tmp = t_1
    else if (y <= 6.5d-297) then
        tmp = t_0
    else if (y <= 9.5d-252) then
        tmp = t_1
    else if ((y <= 56000000000.0d0) .or. (.not. (y <= 5.9d+56)) .and. (y <= 2.35d+113)) then
        tmp = t_0
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((0.1111111111111111 / x));
	double t_1 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -8e-6) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= -1.26e-225) {
		tmp = t_1;
	} else if (y <= 6.5e-297) {
		tmp = t_0;
	} else if (y <= 9.5e-252) {
		tmp = t_1;
	} else if ((y <= 56000000000.0) || (!(y <= 5.9e+56) && (y <= 2.35e+113))) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	t_1 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -8e-6:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= -1.26e-225:
		tmp = t_1
	elif y <= 6.5e-297:
		tmp = t_0
	elif y <= 9.5e-252:
		tmp = t_1
	elif (y <= 56000000000.0) or (not (y <= 5.9e+56) and (y <= 2.35e+113)):
		tmp = t_0
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	t_1 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -8e-6)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= -1.26e-225)
		tmp = t_1;
	elseif (y <= 6.5e-297)
		tmp = t_0;
	elseif (y <= 9.5e-252)
		tmp = t_1;
	elseif ((y <= 56000000000.0) || (!(y <= 5.9e+56) && (y <= 2.35e+113)))
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	t_1 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -8e-6)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= -1.26e-225)
		tmp = t_1;
	elseif (y <= 6.5e-297)
		tmp = t_0;
	elseif (y <= 9.5e-252)
		tmp = t_1;
	elseif ((y <= 56000000000.0) || (~((y <= 5.9e+56)) && (y <= 2.35e+113)))
		tmp = t_0;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -8e-6], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.26e-225], t$95$1, If[LessEqual[y, 6.5e-297], t$95$0, If[LessEqual[y, 9.5e-252], t$95$1, If[Or[LessEqual[y, 56000000000.0], And[N[Not[LessEqual[y, 5.9e+56]], $MachinePrecision], LessEqual[y, 2.35e+113]]], t$95$0, N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.99999999999999964e-6

   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. associate-*l* 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 74.1%

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

      1. *-commutative 74.1%

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

   6. Simplified 74.1%

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

   if -7.99999999999999964e-6 < y < -1.2599999999999999e-225 or 6.5000000000000002e-297 < y < 9.4999999999999993e-252

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. *-commutative 97.3%

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

      2. sub-neg 97.3%

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

      3. metadata-eval 97.3%

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

      4. associate-*r/ 97.5%

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

      5. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in x around inf 61.0%

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

   if -1.2599999999999999e-225 < y < 6.5000000000000002e-297 or 9.4999999999999993e-252 < y < 5.6e10 or 5.9000000000000001e56 < y < 2.3499999999999999e113

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 58.1%

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

      1. expm1-log1p-u 54.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      2. expm1-udef 54.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)} - 1} \]

      3. associate-*r* 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}}\right)} - 1 \]

      4. *-commutative 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      5. metadata-eval 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      6. sqrt-prod 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      7. add-sqr-sqrt 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 9} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      8. metadata-eval 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(-3 \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      9. swap-sqr 54.6%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      10. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\sqrt{x} \cdot -3} \cdot \sqrt{\sqrt{x} \cdot -3}\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      11. add-sqr-sqrt 1.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot -3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   6. Applied egg-rr 1.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 1.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-log1p 2.2%

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

      3. *-commutative 2.2%

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

      4. associate-*r* 2.2%

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

      5. associate-*l/ 2.2%

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

      6. *-commutative 2.2%

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

      7. rem-square-sqrt 2.2%

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

      8. times-frac 2.2%

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

      9. *-inverses 2.2%

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

      10. *-lft-identity 2.2%

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

      11. metadata-eval 2.2%

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

      12. metadata-eval 2.2%

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

      13. *-inverses 2.2%

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

      14. associate-/l* 2.2%

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

      15. associate-*l/ 2.2%

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

      16. associate-*r/ 2.2%

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

      17. associate-*r* 2.2%

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

      18. *-commutative 2.2%

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

      19. associate-*r* 2.2%

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

      20. associate-*r* 2.2%

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

      21. *-commutative 2.2%

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

   8. Simplified 2.2%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 58.4%

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

      3. frac-times 58.4%

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

      4. metadata-eval 58.4%

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

      5. add-sqr-sqrt 58.5%

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

      6. pow1/2 58.5%

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

   10. Applied egg-rr 58.5%

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

       1. unpow1/2 58.5%

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

   12. Simplified 58.5%

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

   if 5.6e10 < y < 5.9000000000000001e56 or 2.3499999999999999e113 < 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%

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

      2. associate-*l* 99.9%

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

      3. associate--l+ 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-lft-in 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-/r* 99.7%

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

      13. associate-*r/ 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 87.8%

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

      1. associate-*r* 87.9%

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

   6. Simplified 87.9%

      \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot \sqrt{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-6}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 56000000000 \lor \neg \left(y \leq 5.9 \cdot 10^{+56}\right) \land y \leq 2.35 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]

Alternative 4: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ t_1 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -8 \cdot 10^{-6}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-297}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 56000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+48} \lor \neg \left(y \leq 2.35 \cdot 10^{+113}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x))) (t_1 (* (sqrt x) -3.0)))
   (if (<= y -8e-6)
     (* 3.0 (* (sqrt x) y))
     (if (<= y -1.26e-225)
       t_1
       (if (<= y 3e-297)
         t_0
         (if (<= y 1.2e-252)
           t_1
           (if (<= y 56000000000.0)
             t_0
             (if (or (<= y 5.7e+48) (not (<= y 2.35e+113)))
               (* (sqrt x) (* 3.0 y))
               (/ (* (sqrt x) 0.3333333333333333) x)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double t_1 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -8e-6) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= -1.26e-225) {
		tmp = t_1;
	} else if (y <= 3e-297) {
		tmp = t_0;
	} else if (y <= 1.2e-252) {
		tmp = t_1;
	} else if (y <= 56000000000.0) {
		tmp = t_0;
	} else if ((y <= 5.7e+48) || !(y <= 2.35e+113)) {
		tmp = sqrt(x) * (3.0 * y);
	} else {
		tmp = (sqrt(x) * 0.3333333333333333) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    t_1 = sqrt(x) * (-3.0d0)
    if (y <= (-8d-6)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= (-1.26d-225)) then
        tmp = t_1
    else if (y <= 3d-297) then
        tmp = t_0
    else if (y <= 1.2d-252) then
        tmp = t_1
    else if (y <= 56000000000.0d0) then
        tmp = t_0
    else if ((y <= 5.7d+48) .or. (.not. (y <= 2.35d+113))) then
        tmp = sqrt(x) * (3.0d0 * y)
    else
        tmp = (sqrt(x) * 0.3333333333333333d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((0.1111111111111111 / x));
	double t_1 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -8e-6) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= -1.26e-225) {
		tmp = t_1;
	} else if (y <= 3e-297) {
		tmp = t_0;
	} else if (y <= 1.2e-252) {
		tmp = t_1;
	} else if (y <= 56000000000.0) {
		tmp = t_0;
	} else if ((y <= 5.7e+48) || !(y <= 2.35e+113)) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else {
		tmp = (Math.sqrt(x) * 0.3333333333333333) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	t_1 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -8e-6:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= -1.26e-225:
		tmp = t_1
	elif y <= 3e-297:
		tmp = t_0
	elif y <= 1.2e-252:
		tmp = t_1
	elif y <= 56000000000.0:
		tmp = t_0
	elif (y <= 5.7e+48) or not (y <= 2.35e+113):
		tmp = math.sqrt(x) * (3.0 * y)
	else:
		tmp = (math.sqrt(x) * 0.3333333333333333) / x
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	t_1 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -8e-6)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= -1.26e-225)
		tmp = t_1;
	elseif (y <= 3e-297)
		tmp = t_0;
	elseif (y <= 1.2e-252)
		tmp = t_1;
	elseif (y <= 56000000000.0)
		tmp = t_0;
	elseif ((y <= 5.7e+48) || !(y <= 2.35e+113))
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	else
		tmp = Float64(Float64(sqrt(x) * 0.3333333333333333) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	t_1 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -8e-6)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= -1.26e-225)
		tmp = t_1;
	elseif (y <= 3e-297)
		tmp = t_0;
	elseif (y <= 1.2e-252)
		tmp = t_1;
	elseif (y <= 56000000000.0)
		tmp = t_0;
	elseif ((y <= 5.7e+48) || ~((y <= 2.35e+113)))
		tmp = sqrt(x) * (3.0 * y);
	else
		tmp = (sqrt(x) * 0.3333333333333333) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -8e-6], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.26e-225], t$95$1, If[LessEqual[y, 3e-297], t$95$0, If[LessEqual[y, 1.2e-252], t$95$1, If[LessEqual[y, 56000000000.0], t$95$0, If[Or[LessEqual[y, 5.7e+48], N[Not[LessEqual[y, 2.35e+113]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -7.99999999999999964e-6

   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. associate-*l* 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 74.1%

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

      1. *-commutative 74.1%

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

   6. Simplified 74.1%

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

   if -7.99999999999999964e-6 < y < -1.2599999999999999e-225 or 2.99999999999999995e-297 < y < 1.2000000000000001e-252

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. *-commutative 97.3%

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

      2. sub-neg 97.3%

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

      3. metadata-eval 97.3%

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

      4. associate-*r/ 97.5%

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

      5. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in x around inf 61.0%

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

   if -1.2599999999999999e-225 < y < 2.99999999999999995e-297 or 1.2000000000000001e-252 < y < 5.6e10

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 56.3%

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

      1. expm1-log1p-u 52.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      2. expm1-udef 53.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)} - 1} \]

      3. associate-*r* 53.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}}\right)} - 1 \]

      4. *-commutative 53.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      5. metadata-eval 53.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      6. sqrt-prod 53.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      7. add-sqr-sqrt 53.0%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 9} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      8. metadata-eval 53.0%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(-3 \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      9. swap-sqr 53.0%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      10. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\sqrt{x} \cdot -3} \cdot \sqrt{\sqrt{x} \cdot -3}\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      11. add-sqr-sqrt 1.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot -3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   6. Applied egg-rr 1.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 1.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-log1p 2.4%

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

      3. *-commutative 2.4%

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

      4. associate-*r* 2.4%

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

      5. associate-*l/ 2.4%

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

      6. *-commutative 2.4%

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

      7. rem-square-sqrt 2.4%

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

      8. times-frac 2.4%

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

      9. *-inverses 2.4%

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

      10. *-lft-identity 2.4%

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

      11. metadata-eval 2.4%

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

      12. metadata-eval 2.4%

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

      13. *-inverses 2.4%

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

      14. associate-/l* 2.4%

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

      15. associate-*l/ 2.4%

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

      16. associate-*r/ 2.4%

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

      17. associate-*r* 2.4%

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

      18. *-commutative 2.4%

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

      19. associate-*r* 2.4%

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

      20. associate-*r* 2.4%

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

      21. *-commutative 2.4%

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

   8. Simplified 2.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 56.5%

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

      3. frac-times 56.6%

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

      4. metadata-eval 56.6%

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

      5. add-sqr-sqrt 56.6%

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

      6. pow1/2 56.6%

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

   10. Applied egg-rr 56.6%

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

       1. unpow1/2 56.6%

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

   12. Simplified 56.6%

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

   if 5.6e10 < y < 5.69999999999999968e48 or 2.3499999999999999e113 < 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%

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

      2. associate-*l* 99.9%

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

      3. associate--l+ 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-lft-in 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-/r* 99.7%

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

      13. associate-*r/ 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 87.8%

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

      1. associate-*r* 87.9%

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

   6. Simplified 87.9%

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

   if 5.69999999999999968e48 < y < 2.3499999999999999e113

   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. associate-*l* 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 65.3%

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

      1. associate-*r/ 65.5%

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

      2. associate-*r/ 65.5%

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

      3. *-commutative 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. associate-*r* 65.7%

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

      2. metadata-eval 65.7%

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

   8. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \sqrt{x}}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-6}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 56000000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+48} \lor \neg \left(y \leq 2.35 \cdot 10^{+113}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\\ \end{array} \]

Alternative 5: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{1}{\frac{x}{0.3333333333333333}} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.45e-55)
   (sqrt (/ 0.1111111111111111 x))
   (if (<= x 2.5e-28)
     (* (sqrt x) (* 3.0 y))
     (if (<= x 4.6e-9)
       (* (sqrt x) (- (/ 1.0 (/ x 0.3333333333333333)) 3.0))
       (* (sqrt x) (* 3.0 (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.45e-55) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (x <= 2.5e-28) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (x <= 4.6e-9) {
		tmp = sqrt(x) * ((1.0 / (x / 0.3333333333333333)) - 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 (x <= 1.45d-55) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (x <= 2.5d-28) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (x <= 4.6d-9) then
        tmp = sqrt(x) * ((1.0d0 / (x / 0.3333333333333333d0)) - 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 (x <= 1.45e-55) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (x <= 2.5e-28) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (x <= 4.6e-9) {
		tmp = Math.sqrt(x) * ((1.0 / (x / 0.3333333333333333)) - 3.0);
	} else {
		tmp = Math.sqrt(x) * (3.0 * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.45e-55:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif x <= 2.5e-28:
		tmp = math.sqrt(x) * (3.0 * y)
	elif x <= 4.6e-9:
		tmp = math.sqrt(x) * ((1.0 / (x / 0.3333333333333333)) - 3.0)
	else:
		tmp = math.sqrt(x) * (3.0 * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.45e-55)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (x <= 2.5e-28)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (x <= 4.6e-9)
		tmp = Float64(sqrt(x) * Float64(Float64(1.0 / Float64(x / 0.3333333333333333)) - 3.0));
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.45e-55)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (x <= 2.5e-28)
		tmp = sqrt(x) * (3.0 * y);
	elseif (x <= 4.6e-9)
		tmp = sqrt(x) * ((1.0 / (x / 0.3333333333333333)) - 3.0);
	else
		tmp = sqrt(x) * (3.0 * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.45e-55], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.5e-28], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e-9], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(1.0 / N[(x / 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.45e-55

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 81.7%

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

      1. expm1-log1p-u 75.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      2. expm1-udef 75.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)} - 1} \]

      3. associate-*r* 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}}\right)} - 1 \]

      4. *-commutative 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      5. metadata-eval 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      6. sqrt-prod 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      7. add-sqr-sqrt 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 9} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      8. metadata-eval 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(-3 \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      9. swap-sqr 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      10. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\sqrt{x} \cdot -3} \cdot \sqrt{\sqrt{x} \cdot -3}\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      11. add-sqr-sqrt 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot -3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-log1p 1.3%

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

      3. *-commutative 1.3%

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

      4. associate-*r* 1.3%

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

      5. associate-*l/ 1.3%

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

      6. *-commutative 1.3%

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

      7. rem-square-sqrt 1.3%

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

      8. times-frac 1.3%

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

      9. *-inverses 1.3%

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

      10. *-lft-identity 1.3%

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

      11. metadata-eval 1.3%

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

      12. metadata-eval 1.3%

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

      13. *-inverses 1.3%

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

      14. associate-/l* 1.3%

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

      15. associate-*l/ 1.3%

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

      16. associate-*r/ 1.3%

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

      17. associate-*r* 1.3%

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

      18. *-commutative 1.3%

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

      19. associate-*r* 1.3%

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

      20. associate-*r* 1.3%

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

      21. *-commutative 1.3%

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

   8. Simplified 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 81.9%

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

      3. frac-times 82.0%

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

      4. metadata-eval 82.0%

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

      5. add-sqr-sqrt 82.1%

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

      6. pow1/2 82.1%

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

   10. Applied egg-rr 82.1%

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

       1. unpow1/2 82.1%

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

   12. Simplified 82.1%

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

   if 1.45e-55 < x < 2.5000000000000001e-28

   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%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 65.1%

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

      1. associate-*r* 65.4%

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

   6. Simplified 65.4%

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

   if 2.5000000000000001e-28 < x < 4.5999999999999998e-9

   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%

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

      2. associate-*l* 99.2%

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

      3. associate--l+ 99.2%

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

      4. distribute-lft-in 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.0%

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

      13. associate-*r/ 99.0%

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

      14. metadata-eval 99.0%

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

      15. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 75.7%

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

      1. div-inv 75.7%

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

      2. clear-num 75.9%

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

   6. Applied egg-rr 75.9%

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

   if 4.5999999999999998e-9 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.7%

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

      3. +-commutative 99.7%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-rgt-in 99.7%

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

      7. *-commutative 99.7%

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

      8. fma-def 99.6%

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

      9. sub-neg 99.6%

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

      10. metadata-eval 99.6%

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

      11. associate-*l/ 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.2%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(y - 1\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{1}{\frac{x}{0.3333333333333333}} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 6: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{--0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6e-55)
   (sqrt (/ 0.1111111111111111 x))
   (if (<= x 9.2e-29)
     (* (sqrt x) (* 3.0 y))
     (if (<= x 3.9e-16)
       (/ (- -0.3333333333333333) (sqrt x))
       (* (sqrt x) (* 3.0 (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6e-55) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (x <= 9.2e-29) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (x <= 3.9e-16) {
		tmp = -(-0.3333333333333333) / sqrt(x);
	} 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 (x <= 6d-55) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (x <= 9.2d-29) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (x <= 3.9d-16) then
        tmp = -(-0.3333333333333333d0) / sqrt(x)
    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 (x <= 6e-55) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (x <= 9.2e-29) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (x <= 3.9e-16) {
		tmp = -(-0.3333333333333333) / Math.sqrt(x);
	} else {
		tmp = Math.sqrt(x) * (3.0 * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6e-55:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif x <= 9.2e-29:
		tmp = math.sqrt(x) * (3.0 * y)
	elif x <= 3.9e-16:
		tmp = -(-0.3333333333333333) / math.sqrt(x)
	else:
		tmp = math.sqrt(x) * (3.0 * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6e-55)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (x <= 9.2e-29)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (x <= 3.9e-16)
		tmp = Float64(Float64(-(-0.3333333333333333)) / sqrt(x));
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6e-55)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (x <= 9.2e-29)
		tmp = sqrt(x) * (3.0 * y);
	elseif (x <= 3.9e-16)
		tmp = -(-0.3333333333333333) / sqrt(x);
	else
		tmp = sqrt(x) * (3.0 * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6e-55], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 9.2e-29], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e-16], N[((--0.3333333333333333) / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.00000000000000033e-55

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 81.7%

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

      1. expm1-log1p-u 75.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      2. expm1-udef 75.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)} - 1} \]

      3. associate-*r* 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}}\right)} - 1 \]

      4. *-commutative 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      5. metadata-eval 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      6. sqrt-prod 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      7. add-sqr-sqrt 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 9} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      8. metadata-eval 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(-3 \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      9. swap-sqr 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      10. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\sqrt{x} \cdot -3} \cdot \sqrt{\sqrt{x} \cdot -3}\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      11. add-sqr-sqrt 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot -3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-log1p 1.3%

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

      3. *-commutative 1.3%

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

      4. associate-*r* 1.3%

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

      5. associate-*l/ 1.3%

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

      6. *-commutative 1.3%

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

      7. rem-square-sqrt 1.3%

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

      8. times-frac 1.3%

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

      9. *-inverses 1.3%

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

      10. *-lft-identity 1.3%

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

      11. metadata-eval 1.3%

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

      12. metadata-eval 1.3%

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

      13. *-inverses 1.3%

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

      14. associate-/l* 1.3%

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

      15. associate-*l/ 1.3%

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

      16. associate-*r/ 1.3%

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

      17. associate-*r* 1.3%

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

      18. *-commutative 1.3%

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

      19. associate-*r* 1.3%

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

      20. associate-*r* 1.3%

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

      21. *-commutative 1.3%

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

   8. Simplified 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 81.9%

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

      3. frac-times 82.0%

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

      4. metadata-eval 82.0%

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

      5. add-sqr-sqrt 82.1%

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

      6. pow1/2 82.1%

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

   10. Applied egg-rr 82.1%

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

       1. unpow1/2 82.1%

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

   12. Simplified 82.1%

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

   if 6.00000000000000033e-55 < x < 9.19999999999999965e-29

   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%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 65.1%

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

      1. associate-*r* 65.4%

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

   6. Simplified 65.4%

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

   if 9.19999999999999965e-29 < x < 3.89999999999999977e-16

   1. Initial program 99.2%

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

      1. associate-*l* 99.0%

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

      2. associate--l+ 99.0%

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

      3. sub-neg 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-/r* 98.7%

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

      6. metadata-eval 98.7%

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

      7. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in x around 0 81.3%

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

      1. associate-*r/ 81.0%

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

      2. associate-*r/ 81.5%

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

      3. *-commutative 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. *-commutative 81.5%

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

      2. associate-*l* 81.3%

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

      3. metadata-eval 81.3%

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

      4. distribute-rgt-neg-in 81.3%

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

      5. distribute-rgt-neg-out 81.3%

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

      6. distribute-neg-frac 81.3%

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

      7. *-commutative 81.3%

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

      8. associate-*r/ 81.6%

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

      9. *-commutative 81.6%

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

      10. associate-*r* 81.3%

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

   8. Simplified 81.6%

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

   if 3.89999999999999977e-16 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.7%

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

      3. +-commutative 99.7%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-rgt-in 99.7%

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

      7. *-commutative 99.7%

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

      8. fma-def 99.6%

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

      9. sub-neg 99.6%

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

      10. metadata-eval 99.6%

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

      11. associate-*l/ 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 97.4%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(y - 1\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{--0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 7: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.2e-54)
   (sqrt (/ 0.1111111111111111 x))
   (if (<= x 5.1e-29)
     (* (sqrt x) (* 3.0 y))
     (if (<= x 3.1e-9)
       (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
       (* (sqrt x) (* 3.0 (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.2e-54) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (x <= 5.1e-29) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (x <= 3.1e-9) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} 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 (x <= 2.2d-54) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (x <= 5.1d-29) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (x <= 3.1d-9) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    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 (x <= 2.2e-54) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (x <= 5.1e-29) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (x <= 3.1e-9) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt(x) * (3.0 * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.2e-54:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif x <= 5.1e-29:
		tmp = math.sqrt(x) * (3.0 * y)
	elif x <= 3.1e-9:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = math.sqrt(x) * (3.0 * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.2e-54)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (x <= 5.1e-29)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (x <= 3.1e-9)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.2e-54)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (x <= 5.1e-29)
		tmp = sqrt(x) * (3.0 * y);
	elseif (x <= 3.1e-9)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = sqrt(x) * (3.0 * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.2e-54], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.1e-29], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e-9], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.2e-54

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 81.7%

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

      1. expm1-log1p-u 75.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      2. expm1-udef 75.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)} - 1} \]

      3. associate-*r* 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}}\right)} - 1 \]

      4. *-commutative 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      5. metadata-eval 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      6. sqrt-prod 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      7. add-sqr-sqrt 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 9} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      8. metadata-eval 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(-3 \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      9. swap-sqr 75.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      10. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\sqrt{x} \cdot -3} \cdot \sqrt{\sqrt{x} \cdot -3}\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      11. add-sqr-sqrt 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot -3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-log1p 1.3%

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

      3. *-commutative 1.3%

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

      4. associate-*r* 1.3%

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

      5. associate-*l/ 1.3%

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

      6. *-commutative 1.3%

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

      7. rem-square-sqrt 1.3%

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

      8. times-frac 1.3%

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

      9. *-inverses 1.3%

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

      10. *-lft-identity 1.3%

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

      11. metadata-eval 1.3%

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

      12. metadata-eval 1.3%

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

      13. *-inverses 1.3%

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

      14. associate-/l* 1.3%

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

      15. associate-*l/ 1.3%

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

      16. associate-*r/ 1.3%

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

      17. associate-*r* 1.3%

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

      18. *-commutative 1.3%

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

      19. associate-*r* 1.3%

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

      20. associate-*r* 1.3%

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

      21. *-commutative 1.3%

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

   8. Simplified 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 81.9%

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

      3. frac-times 82.0%

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

      4. metadata-eval 82.0%

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

      5. add-sqr-sqrt 82.1%

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

      6. pow1/2 82.1%

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

   10. Applied egg-rr 82.1%

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

       1. unpow1/2 82.1%

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

   12. Simplified 82.1%

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

   if 2.2e-54 < x < 5.09999999999999986e-29

   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%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 65.1%

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

      1. associate-*r* 65.4%

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

   6. Simplified 65.4%

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

   if 5.09999999999999986e-29 < x < 3.10000000000000005e-9

   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%

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

      2. associate-*l* 99.2%

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

      3. associate--l+ 99.2%

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

      4. distribute-lft-in 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.0%

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

      13. associate-*r/ 99.0%

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

      14. metadata-eval 99.0%

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

      15. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. sub-neg 75.7%

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

      3. metadata-eval 75.7%

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

      4. associate-*r/ 75.7%

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

      5. metadata-eval 75.7%

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

   6. Simplified 75.7%

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

   if 3.10000000000000005e-9 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.7%

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

      3. +-commutative 99.7%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-rgt-in 99.7%

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

      7. *-commutative 99.7%

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

      8. fma-def 99.6%

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

      9. sub-neg 99.6%

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

      10. metadata-eval 99.6%

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

      11. associate-*l/ 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.2%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(y - 1\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 3.0 * (sqrt(x) * (y + ((0.1111111111111111 / x) + -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 + ((0.1111111111111111d0 / x) + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $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. associate-*l* 99.5%

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

   2. associate--l+ 99.5%

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

   3. sub-neg 99.5%

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

   4. *-commutative 99.5%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 9: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * N[(N[(0.1111111111111111 / x), $MachinePrecision] + N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $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. associate-*l* 99.5%

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

   2. associate--l+ 99.5%

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

   3. sub-neg 99.5%

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

   4. *-commutative 99.5%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in y around 0 99.4%

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

   1. distribute-lft-out 99.4%

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

   2. +-commutative 99.4%

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

   3. distribute-rgt-out 99.4%

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

   4. sub-neg 99.4%

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

   5. associate-*r/ 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. +-commutative 99.4%

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

   9. *-commutative 99.4%

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

   10. associate-*l* 99.5%

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

   11. +-commutative 99.5%

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

   12. associate-+r+ 99.5%

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

   13. +-commutative 99.5%

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

   14. associate-+l+ 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 10: 61.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.68) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.68) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.68:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.68)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.68], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 72.0%

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

      1. expm1-log1p-u 67.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      2. expm1-udef 67.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)} - 1} \]

      3. associate-*r* 67.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}}\right)} - 1 \]

      4. *-commutative 67.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      5. metadata-eval 67.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      6. sqrt-prod 67.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      7. add-sqr-sqrt 67.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 9} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      8. metadata-eval 67.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(-3 \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      9. swap-sqr 67.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      10. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\sqrt{x} \cdot -3} \cdot \sqrt{\sqrt{x} \cdot -3}\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

      11. add-sqr-sqrt 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot -3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-log1p 1.7%

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

      3. *-commutative 1.7%

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

      4. associate-*r* 1.7%

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

      5. associate-*l/ 1.7%

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

      6. *-commutative 1.7%

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

      7. rem-square-sqrt 1.7%

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

      8. times-frac 1.7%

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

      9. *-inverses 1.7%

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

      10. *-lft-identity 1.7%

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

      11. metadata-eval 1.7%

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

      12. metadata-eval 1.7%

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

      13. *-inverses 1.7%

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

      14. associate-/l* 1.7%

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

      15. associate-*l/ 1.7%

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

      16. associate-*r/ 1.7%

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

      17. associate-*r* 1.7%

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

      18. *-commutative 1.7%

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

      19. associate-*r* 1.7%

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

      20. associate-*r* 1.7%

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

      21. *-commutative 1.7%

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

   8. Simplified 1.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 72.2%

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

      3. frac-times 72.2%

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

      4. metadata-eval 72.2%

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

      5. add-sqr-sqrt 72.3%

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

      6. pow1/2 72.3%

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

   10. Applied egg-rr 72.3%

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

       1. unpow1/2 72.3%

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

   12. Simplified 72.3%

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

   if 0.680000000000000049 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.7%

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

      3. associate--l+ 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-lft-in 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-/r* 99.7%

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

      13. associate-*r/ 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 47.3%

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

      1. *-commutative 47.3%

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

      2. sub-neg 47.3%

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

      3. metadata-eval 47.3%

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

      4. associate-*r/ 47.3%

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

      5. metadata-eval 47.3%

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

   6. Simplified 47.3%

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

   7. Taylor expanded in x around inf 47.3%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 11: 37.9% 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. associate-*l* 99.5%

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

   2. associate--l+ 99.5%

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

   3. sub-neg 99.5%

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

   4. *-commutative 99.5%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 36.4%

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

   1. expm1-log1p-u 34.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)\right)} \]

   2. expm1-udef 34.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3 \cdot \left(\sqrt{x} \cdot \frac{0.1111111111111111}{x}\right)\right)} - 1} \]

   3. associate-*r* 34.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}}\right)} - 1 \]

   4. *-commutative 34.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   5. metadata-eval 34.3%

      \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   6. sqrt-prod 34.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   7. add-sqr-sqrt 34.3%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 9} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   8. metadata-eval 34.3%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(-3 \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   9. swap-sqr 34.3%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   10. sqrt-unprod 0.0%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\sqrt{x} \cdot -3} \cdot \sqrt{\sqrt{x} \cdot -3}\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

   11. add-sqr-sqrt 1.3%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot -3\right)} \cdot \frac{0.1111111111111111}{x}\right)} - 1 \]

6. Applied egg-rr 1.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 1.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{x} \cdot -3\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

   2. expm1-log1p 2.7%

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

   3. *-commutative 2.7%

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

   4. associate-*r* 2.7%

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

   5. associate-*l/ 2.7%

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

   6. *-commutative 2.7%

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

   7. rem-square-sqrt 2.7%

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

   8. times-frac 2.7%

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

   9. *-inverses 2.7%

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

   10. *-lft-identity 2.7%

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

   11. metadata-eval 2.7%

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

   12. metadata-eval 2.7%

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

   13. *-inverses 2.7%

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

   14. associate-/l* 2.7%

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

   15. associate-*l/ 2.7%

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

   16. associate-*r/ 2.7%

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

   17. associate-*r* 2.7%

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

   18. *-commutative 2.7%

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

   19. associate-*r* 2.7%

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

   20. associate-*r* 2.7%

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

   21. *-commutative 2.7%

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

8. Simplified 2.7%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 36.5%

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

   3. frac-times 36.5%

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

   4. metadata-eval 36.5%

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

   5. add-sqr-sqrt 36.6%

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

   6. pow1/2 36.6%

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

10. Applied egg-rr 36.6%

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

    1. unpow1/2 36.6%

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

12. Simplified 36.6%

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

13. Final simplification 36.6%

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

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 2023274 
(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)))