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

Percentage Accurate: 99.4% → 99.4%

Time: 9.5s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

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]

Derivation

1. Initial program 99.5%

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

2. Final simplification 99.5%

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

Alternative 2: 60.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_1 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;y \leq -1.22 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.85 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (* (sqrt x) y))) (t_1 (sqrt (/ 0.1111111111111111 x))))
   (if (<= y -1.22e+83)
     t_0
     (if (<= y -1.35e-148)
       t_1
       (if (<= y -3.85e-208)
         (* (sqrt x) -3.0)
         (if (<= y -1.05e-307)
           t_1
           (if (<= y 1.0) (- (sqrt (* x 9.0))) t_0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (sqrt(x) * y);
	double t_1 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -1.22e+83) {
		tmp = t_0;
	} else if (y <= -1.35e-148) {
		tmp = t_1;
	} else if (y <= -3.85e-208) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= -1.05e-307) {
		tmp = t_1;
	} else if (y <= 1.0) {
		tmp = -sqrt((x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 * (sqrt(x) * y)
    t_1 = sqrt((0.1111111111111111d0 / x))
    if (y <= (-1.22d+83)) then
        tmp = t_0
    else if (y <= (-1.35d-148)) then
        tmp = t_1
    else if (y <= (-3.85d-208)) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= (-1.05d-307)) then
        tmp = t_1
    else if (y <= 1.0d0) then
        tmp = -sqrt((x * 9.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * (Math.sqrt(x) * y);
	double t_1 = Math.sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -1.22e+83) {
		tmp = t_0;
	} else if (y <= -1.35e-148) {
		tmp = t_1;
	} else if (y <= -3.85e-208) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= -1.05e-307) {
		tmp = t_1;
	} else if (y <= 1.0) {
		tmp = -Math.sqrt((x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * (math.sqrt(x) * y)
	t_1 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if y <= -1.22e+83:
		tmp = t_0
	elif y <= -1.35e-148:
		tmp = t_1
	elif y <= -3.85e-208:
		tmp = math.sqrt(x) * -3.0
	elif y <= -1.05e-307:
		tmp = t_1
	elif y <= 1.0:
		tmp = -math.sqrt((x * 9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(sqrt(x) * y))
	t_1 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -1.22e+83)
		tmp = t_0;
	elseif (y <= -1.35e-148)
		tmp = t_1;
	elseif (y <= -3.85e-208)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= -1.05e-307)
		tmp = t_1;
	elseif (y <= 1.0)
		tmp = Float64(-sqrt(Float64(x * 9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * (sqrt(x) * y);
	t_1 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (y <= -1.22e+83)
		tmp = t_0;
	elseif (y <= -1.35e-148)
		tmp = t_1;
	elseif (y <= -3.85e-208)
		tmp = sqrt(x) * -3.0;
	elseif (y <= -1.05e-307)
		tmp = t_1;
	elseif (y <= 1.0)
		tmp = -sqrt((x * 9.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.22e+83], t$95$0, If[LessEqual[y, -1.35e-148], t$95$1, If[LessEqual[y, -3.85e-208], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, -1.05e-307], t$95$1, If[LessEqual[y, 1.0], (-N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.22e83 or 1 < 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. Taylor expanded in y around inf 85.3%

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

   if -1.22e83 < y < -1.34999999999999994e-148 or -3.84999999999999986e-208 < y < -1.0500000000000001e-307

   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-/r* 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-/r* 99.4%

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

      4. inv-pow 99.4%

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

      5. *-commutative 99.4%

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

      6. unpow-prod-down 99.2%

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

      7. metadata-eval 99.2%

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

      8. inv-pow 99.2%

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

      9. div-inv 99.3%

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

      10. associate-+r- 99.3%

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

      11. add-sqr-sqrt 60.6%

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

      12. pow2 60.6%

          \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{\frac{0.1111111111111111}{x} + \left(y - 1\right)}\right)}^{2}} \]

   3. Applied egg-rr 60.6%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 60.3%

         \[\leadsto \color{blue}{\sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \cdot \sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}}} \]

      2. sqrt-unprod 60.6%

         \[\leadsto \color{blue}{\sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)}} \]

      3. unpow2 60.6%

         \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)} \cdot \sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      4. add-sqr-sqrt 60.7%

         \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      5. *-commutative 60.7%

         \[\leadsto \sqrt{\color{blue}{\left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      6. unpow2 60.7%

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

      7. add-sqr-sqrt 61.2%

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

      8. *-commutative 61.2%

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

      9. swap-sqr 38.5%

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

   5. Applied egg-rr 38.5%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{0.1111111111111111}{x} + \left(-1 + y\right)\right)}^{2} \cdot \left(x \cdot 9\right)}} \]

   6. Taylor expanded in x around 0 34.1%

      \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{{x}^{2}}} \cdot \left(x \cdot 9\right)} \]
   7. Step-by-step derivation

      1. unpow2 34.1%

         \[\leadsto \sqrt{\frac{0.012345679012345678}{\color{blue}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

   8. Simplified 34.1%

      \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

   9. Taylor expanded in x around 0 56.9%

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

   if -1.34999999999999994e-148 < y < -3.84999999999999986e-208

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 75.3%

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

   3. Taylor expanded in y around 0 75.3%

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

      1. *-commutative 75.3%

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

   5. Simplified 75.3%

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

   if -1.0500000000000001e-307 < y < 1

   1. Initial program 99.5%

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

   2. Taylor expanded in x around inf 60.2%

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

   3. Taylor expanded in y around 0 59.5%

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

      1. *-commutative 59.5%

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

   5. Simplified 59.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 3.2%

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

      3. pow2 3.2%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{x} \cdot -3\right)}^{2}}} \]

   7. Applied egg-rr 3.2%

      \[\leadsto \color{blue}{\sqrt{{\left(\sqrt{x} \cdot -3\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 3.2%

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

      2. swap-sqr 3.2%

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

      3. rem-square-sqrt 3.2%

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

      4. metadata-eval 3.2%

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

   9. Simplified 3.2%

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

       1. sqrt-prod 3.2%

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

       2. metadata-eval 3.2%

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

       3. metadata-eval 3.2%

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

       4. distribute-rgt-neg-in 3.2%

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

       5. add-cbrt-cube 3.2%

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

       6. metadata-eval 3.2%

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

       7. metadata-eval 3.2%

          \[\leadsto -\sqrt{x} \cdot \sqrt[3]{\color{blue}{-27}} \]

       8. add-sqr-sqrt 0.0%

          \[\leadsto -\color{blue}{\sqrt{\sqrt{x} \cdot \sqrt[3]{-27}} \cdot \sqrt{\sqrt{x} \cdot \sqrt[3]{-27}}} \]

       9. distribute-rgt-neg-in 0.0%

          \[\leadsto \color{blue}{\sqrt{\sqrt{x} \cdot \sqrt[3]{-27}} \cdot \left(-\sqrt{\sqrt{x} \cdot \sqrt[3]{-27}}\right)} \]

   11. Applied egg-rr 59.3%

       \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.25} \cdot \left(-{\left(x \cdot 9\right)}^{0.25}\right)} \]
   12. Step-by-step derivation

       1. distribute-rgt-neg-out 59.3%

          \[\leadsto \color{blue}{-{\left(x \cdot 9\right)}^{0.25} \cdot {\left(x \cdot 9\right)}^{0.25}} \]

       2. pow-sqr 59.5%

          \[\leadsto -\color{blue}{{\left(x \cdot 9\right)}^{\left(2 \cdot 0.25\right)}} \]

       3. metadata-eval 59.5%

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

       4. unpow1/2 59.5%

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

   13. Simplified 59.5%

       \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq -3.85 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 3: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x))))
   (if (<= y -1.2e+83)
     (* 3.0 (* (sqrt x) y))
     (if (<= y -9.2e-149)
       t_0
       (if (<= y -4e-205)
         (* (sqrt x) -3.0)
         (if (<= y -8.2e-308)
           t_0
           (if (<= y 1.0) (- (sqrt (* x 9.0))) (* (* 3.0 (sqrt x)) y))))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -1.2e+83) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= -9.2e-149) {
		tmp = t_0;
	} else if (y <= -4e-205) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= -8.2e-308) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = -sqrt((x * 9.0));
	} else {
		tmp = (3.0 * sqrt(x)) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    if (y <= (-1.2d+83)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= (-9.2d-149)) then
        tmp = t_0
    else if (y <= (-4d-205)) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= (-8.2d-308)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = -sqrt((x * 9.0d0))
    else
        tmp = (3.0d0 * sqrt(x)) * 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 tmp;
	if (y <= -1.2e+83) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= -9.2e-149) {
		tmp = t_0;
	} else if (y <= -4e-205) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= -8.2e-308) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = -Math.sqrt((x * 9.0));
	} else {
		tmp = (3.0 * Math.sqrt(x)) * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if y <= -1.2e+83:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= -9.2e-149:
		tmp = t_0
	elif y <= -4e-205:
		tmp = math.sqrt(x) * -3.0
	elif y <= -8.2e-308:
		tmp = t_0
	elif y <= 1.0:
		tmp = -math.sqrt((x * 9.0))
	else:
		tmp = (3.0 * math.sqrt(x)) * y
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -1.2e+83)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= -9.2e-149)
		tmp = t_0;
	elseif (y <= -4e-205)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= -8.2e-308)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(-sqrt(Float64(x * 9.0)));
	else
		tmp = Float64(Float64(3.0 * sqrt(x)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (y <= -1.2e+83)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= -9.2e-149)
		tmp = t_0;
	elseif (y <= -4e-205)
		tmp = sqrt(x) * -3.0;
	elseif (y <= -8.2e-308)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = -sqrt((x * 9.0));
	else
		tmp = (3.0 * sqrt(x)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.2e+83], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -9.2e-149], t$95$0, If[LessEqual[y, -4e-205], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, -8.2e-308], t$95$0, If[LessEqual[y, 1.0], (-N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.19999999999999996e83

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 93.0%

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

   if -1.19999999999999996e83 < y < -9.1999999999999999e-149 or -4e-205 < y < -8.19999999999999965e-308

   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-/r* 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-/r* 99.4%

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

      4. inv-pow 99.4%

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

      5. *-commutative 99.4%

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

      6. unpow-prod-down 99.2%

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

      7. metadata-eval 99.2%

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

      8. inv-pow 99.2%

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

      9. div-inv 99.3%

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

      10. associate-+r- 99.3%

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

      11. add-sqr-sqrt 60.6%

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

      12. pow2 60.6%

          \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{\frac{0.1111111111111111}{x} + \left(y - 1\right)}\right)}^{2}} \]

   3. Applied egg-rr 60.6%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 60.3%

         \[\leadsto \color{blue}{\sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \cdot \sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}}} \]

      2. sqrt-unprod 60.6%

         \[\leadsto \color{blue}{\sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)}} \]

      3. unpow2 60.6%

         \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)} \cdot \sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      4. add-sqr-sqrt 60.7%

         \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      5. *-commutative 60.7%

         \[\leadsto \sqrt{\color{blue}{\left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      6. unpow2 60.7%

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

      7. add-sqr-sqrt 61.2%

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

      8. *-commutative 61.2%

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

      9. swap-sqr 38.5%

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

   5. Applied egg-rr 38.5%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{0.1111111111111111}{x} + \left(-1 + y\right)\right)}^{2} \cdot \left(x \cdot 9\right)}} \]

   6. Taylor expanded in x around 0 34.1%

      \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{{x}^{2}}} \cdot \left(x \cdot 9\right)} \]
   7. Step-by-step derivation

      1. unpow2 34.1%

         \[\leadsto \sqrt{\frac{0.012345679012345678}{\color{blue}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

   8. Simplified 34.1%

      \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

   9. Taylor expanded in x around 0 56.9%

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

   if -9.1999999999999999e-149 < y < -4e-205

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 75.3%

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

   3. Taylor expanded in y around 0 75.3%

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

      1. *-commutative 75.3%

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

   5. Simplified 75.3%

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

   if -8.19999999999999965e-308 < y < 1

   1. Initial program 99.5%

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

   2. Taylor expanded in x around inf 60.2%

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

   3. Taylor expanded in y around 0 59.5%

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

      1. *-commutative 59.5%

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

   5. Simplified 59.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 3.2%

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

      3. pow2 3.2%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{x} \cdot -3\right)}^{2}}} \]

   7. Applied egg-rr 3.2%

      \[\leadsto \color{blue}{\sqrt{{\left(\sqrt{x} \cdot -3\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 3.2%

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

      2. swap-sqr 3.2%

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

      3. rem-square-sqrt 3.2%

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

      4. metadata-eval 3.2%

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

   9. Simplified 3.2%

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

       1. sqrt-prod 3.2%

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

       2. metadata-eval 3.2%

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

       3. metadata-eval 3.2%

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

       4. distribute-rgt-neg-in 3.2%

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

       5. add-cbrt-cube 3.2%

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

       6. metadata-eval 3.2%

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

       7. metadata-eval 3.2%

          \[\leadsto -\sqrt{x} \cdot \sqrt[3]{\color{blue}{-27}} \]

       8. add-sqr-sqrt 0.0%

          \[\leadsto -\color{blue}{\sqrt{\sqrt{x} \cdot \sqrt[3]{-27}} \cdot \sqrt{\sqrt{x} \cdot \sqrt[3]{-27}}} \]

       9. distribute-rgt-neg-in 0.0%

          \[\leadsto \color{blue}{\sqrt{\sqrt{x} \cdot \sqrt[3]{-27}} \cdot \left(-\sqrt{\sqrt{x} \cdot \sqrt[3]{-27}}\right)} \]

   11. Applied egg-rr 59.3%

       \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.25} \cdot \left(-{\left(x \cdot 9\right)}^{0.25}\right)} \]
   12. Step-by-step derivation

       1. distribute-rgt-neg-out 59.3%

          \[\leadsto \color{blue}{-{\left(x \cdot 9\right)}^{0.25} \cdot {\left(x \cdot 9\right)}^{0.25}} \]

       2. pow-sqr 59.5%

          \[\leadsto -\color{blue}{{\left(x \cdot 9\right)}^{\left(2 \cdot 0.25\right)}} \]

       3. metadata-eval 59.5%

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

       4. unpow1/2 59.5%

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

   13. Simplified 59.5%

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

   if 1 < 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. Taylor expanded in y around inf 79.2%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{y} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]

Alternative 4: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;x \leq 4 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 0.056:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x))))
   (if (<= x 4e-73)
     t_0
     (if (<= x 2.9e-24)
       (* 3.0 (* (sqrt x) y))
       (if (<= x 0.056) t_0 (* 3.0 (* (sqrt x) (+ y -1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (x <= 4e-73) {
		tmp = t_0;
	} else if (x <= 2.9e-24) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (x <= 0.056) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    if (x <= 4d-73) then
        tmp = t_0
    else if (x <= 2.9d-24) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (x <= 0.056d0) then
        tmp = t_0
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((0.1111111111111111 / x));
	double tmp;
	if (x <= 4e-73) {
		tmp = t_0;
	} else if (x <= 2.9e-24) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (x <= 0.056) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if x <= 4e-73:
		tmp = t_0
	elif x <= 2.9e-24:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif x <= 0.056:
		tmp = t_0
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (x <= 4e-73)
		tmp = t_0;
	elseif (x <= 2.9e-24)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (x <= 0.056)
		tmp = t_0;
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (x <= 4e-73)
		tmp = t_0;
	elseif (x <= 2.9e-24)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (x <= 0.056)
		tmp = t_0;
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4e-73], t$95$0, If[LessEqual[x, 2.9e-24], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.056], t$95$0, N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.99999999999999999e-73 or 2.8999999999999999e-24 < x < 0.0560000000000000012

   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-/r* 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-/r* 99.4%

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

      4. inv-pow 99.4%

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

      5. *-commutative 99.4%

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

      6. unpow-prod-down 99.2%

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

      7. metadata-eval 99.2%

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

      8. inv-pow 99.2%

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

      9. div-inv 99.3%

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

      10. associate-+r- 99.3%

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

      11. add-sqr-sqrt 84.8%

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

      12. pow2 84.8%

          \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{\frac{0.1111111111111111}{x} + \left(y - 1\right)}\right)}^{2}} \]

   3. Applied egg-rr 84.8%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 84.4%

         \[\leadsto \color{blue}{\sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \cdot \sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}}} \]

      2. sqrt-unprod 82.1%

         \[\leadsto \color{blue}{\sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)}} \]

      3. unpow2 82.1%

         \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)} \cdot \sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      4. add-sqr-sqrt 82.3%

         \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      5. *-commutative 82.3%

         \[\leadsto \sqrt{\color{blue}{\left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      6. unpow2 82.3%

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

      7. add-sqr-sqrt 82.4%

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

      8. *-commutative 82.4%

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

      9. swap-sqr 41.2%

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

   5. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{0.1111111111111111}{x} + \left(-1 + y\right)\right)}^{2} \cdot \left(x \cdot 9\right)}} \]

   6. Taylor expanded in x around 0 34.2%

      \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{{x}^{2}}} \cdot \left(x \cdot 9\right)} \]
   7. Step-by-step derivation

      1. unpow2 34.2%

         \[\leadsto \sqrt{\frac{0.012345679012345678}{\color{blue}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

   8. Simplified 34.2%

      \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

   9. Taylor expanded in x around 0 70.3%

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

   if 3.99999999999999999e-73 < x < 2.8999999999999999e-24

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 76.3%

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

   if 0.0560000000000000012 < x

   1. Initial program 99.7%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-/r* 99.7%

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

      4. inv-pow 99.7%

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

      5. *-commutative 99.7%

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

      6. unpow-prod-down 99.7%

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

      7. metadata-eval 99.7%

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

      8. inv-pow 99.7%

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

      9. div-inv 99.7%

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

      10. associate-+r- 99.7%

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

      11. add-sqr-sqrt 24.2%

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

      12. pow2 24.2%

          \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{\frac{0.1111111111111111}{x} + \left(y - 1\right)}\right)}^{2}} \]

   3. Applied egg-rr 24.2%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \]

   4. Taylor expanded in y around 0 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. *-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. associate-+r+ 99.6%

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

      10. associate-+r+ 99.6%

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

      11. +-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. remove-double-neg 99.6%

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

      14. sub-neg 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 98.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 0.056:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

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(y + Float64(Float64(Float64(1.0 / 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[(y + N[(N[(N[(1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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+ 99.5%

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

   2. associate-/r* 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 7200000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+83)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 7200000000000.0)
     (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
     (* 3.0 (* (sqrt x) (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+83) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 7200000000000.0) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.2d+83)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 7200000000000.0d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+83) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 7200000000000.0) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e+83:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 7200000000000.0:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+83)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 7200000000000.0)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e+83)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 7200000000000.0)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e+83], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7200000000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.19999999999999996e83

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 93.0%

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

   if -1.19999999999999996e83 < y < 7.2e12

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0 91.4%

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

      1. associate-*r* 91.4%

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

      2. *-commutative 91.4%

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

      3. associate-*r/ 91.4%

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

      4. metadata-eval 91.4%

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

      5. sub-neg 91.4%

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

      6. metadata-eval 91.4%

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

      7. distribute-rgt-in 91.4%

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

      8. associate-*l/ 91.4%

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

      9. metadata-eval 91.4%

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

      10. metadata-eval 91.4%

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

   4. Simplified 91.4%

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

   if 7.2e12 < 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. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-/r* 99.6%

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

      4. inv-pow 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow-prod-down 99.6%

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

      7. metadata-eval 99.6%

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

      8. inv-pow 99.6%

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

      9. div-inv 99.6%

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

      10. associate-+r- 99.6%

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

      11. add-sqr-sqrt 99.1%

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

      12. pow2 99.1%

          \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{\frac{0.1111111111111111}{x} + \left(y - 1\right)}\right)}^{2}} \]

   3. Applied egg-rr 99.1%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \]

   4. Taylor expanded in y around 0 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. *-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. associate-*r/ 99.5%

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

      5. metadata-eval 99.5%

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

      6. metadata-eval 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-lft-in 99.4%

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

      9. associate-+r+ 99.4%

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

      10. associate-+r+ 99.4%

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

      11. +-commutative 99.4%

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

      12. +-commutative 99.4%

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

      13. remove-double-neg 99.4%

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

      14. sub-neg 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around inf 81.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 7200000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 7: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 9200000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+83)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 9200000000000.0)
     (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
     (* (* 3.0 (sqrt x)) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+83) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 9200000000000.0) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.2d+83)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 9200000000000.0d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = (3.0d0 * sqrt(x)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+83) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 9200000000000.0) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = (3.0 * Math.sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e+83:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 9200000000000.0:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = (3.0 * math.sqrt(x)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+83)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 9200000000000.0)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(Float64(3.0 * sqrt(x)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e+83)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 9200000000000.0)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e+83], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9200000000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.19999999999999996e83

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 93.0%

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

   if -1.19999999999999996e83 < y < 9.2e12

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0 91.4%

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

      1. associate-*r* 91.4%

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

      2. *-commutative 91.4%

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

      3. associate-*r/ 91.4%

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

      4. metadata-eval 91.4%

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

      5. sub-neg 91.4%

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

      6. metadata-eval 91.4%

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

      7. distribute-rgt-in 91.4%

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

      8. associate-*l/ 91.4%

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

      9. metadata-eval 91.4%

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

      10. metadata-eval 91.4%

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

   4. Simplified 91.4%

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

   if 9.2e12 < 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. Taylor expanded in x around inf 81.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 9200000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 8: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + \frac{0.1111111111111111}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112)
   (* 3.0 (* (sqrt x) (+ y (/ 0.1111111111111111 x))))
   (* (* 3.0 (sqrt x)) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = 3.0 * (sqrt(x) * (y + (0.1111111111111111 / x)));
	} else {
		tmp = (3.0 * sqrt(x)) * (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 <= 0.112d0) then
        tmp = 3.0d0 * (sqrt(x) * (y + (0.1111111111111111d0 / x)))
    else
        tmp = (3.0d0 * sqrt(x)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = 3.0 * (Math.sqrt(x) * (y + (0.1111111111111111 / x)));
	} else {
		tmp = (3.0 * Math.sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = 3.0 * (math.sqrt(x) * (y + (0.1111111111111111 / x)))
	else:
		tmp = (3.0 * math.sqrt(x)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + Float64(0.1111111111111111 / x))));
	else
		tmp = Float64(Float64(3.0 * sqrt(x)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = 3.0 * (sqrt(x) * (y + (0.1111111111111111 / x)));
	else
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   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-/r* 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-/r* 99.4%

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

      4. inv-pow 99.4%

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

      5. *-commutative 99.4%

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

      6. unpow-prod-down 99.3%

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

      7. metadata-eval 99.3%

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

      8. inv-pow 99.3%

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

      9. div-inv 99.3%

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

      10. associate-+r- 99.3%

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

      11. add-sqr-sqrt 79.9%

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

      12. pow2 79.9%

          \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{\frac{0.1111111111111111}{x} + \left(y - 1\right)}\right)}^{2}} \]

   3. Applied egg-rr 79.9%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \]

   4. Taylor expanded in y around 0 99.3%

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

      1. distribute-lft-out 99.3%

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

      2. *-commutative 99.3%

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

      3. sub-neg 99.3%

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

      4. associate-*r/ 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

      7. *-commutative 99.4%

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

      8. distribute-lft-in 99.3%

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

      9. associate-+r+ 99.3%

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

      10. associate-+r+ 99.3%

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

      11. +-commutative 99.3%

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

      12. +-commutative 99.3%

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

      13. remove-double-neg 99.3%

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

      14. sub-neg 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around 0 96.5%

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

   if 0.112000000000000002 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + \frac{0.1111111111111111}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 9: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) - ((-0.1111111111111111d0) / x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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-/r* 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-/r* 99.5%

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

   4. inv-pow 99.5%

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

   5. *-commutative 99.5%

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

   6. unpow-prod-down 99.5%

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

   7. metadata-eval 99.5%

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

   8. inv-pow 99.5%

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

   9. div-inv 99.5%

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

   10. associate-+r- 99.5%

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

   11. add-sqr-sqrt 52.3%

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

   12. pow2 52.3%

       \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{\frac{0.1111111111111111}{x} + \left(y - 1\right)}\right)}^{2}} \]

3. Applied egg-rr 52.3%

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \]

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 99.5%

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

   2. *-commutative 99.5%

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

   3. sub-neg 99.5%

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

   4. associate-*r/ 99.5%

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

   5. metadata-eval 99.5%

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

   6. metadata-eval 99.5%

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

   7. *-commutative 99.5%

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

   8. distribute-lft-in 99.5%

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

   9. associate-+r+ 99.5%

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

   10. associate-+r+ 99.5%

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

   11. +-commutative 99.5%

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

   12. +-commutative 99.5%

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

   13. remove-double-neg 99.5%

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

   14. sub-neg 99.5%

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

   15. distribute-neg-frac 99.5%

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

   16. metadata-eval 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 10: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate--l+ 99.5%

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

   3. distribute-rgt-in 99.5%

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

   4. remove-double-neg 99.5%

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

   5. distribute-lft-neg-in 99.5%

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

   6. distribute-rgt-neg-in 99.5%

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

   7. mul-1-neg 99.5%

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

   8. metadata-eval 99.5%

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

   9. *-commutative 99.5%

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

   10. associate-/r* 99.5%

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

   11. distribute-neg-frac 99.5%

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

   12. *-commutative 99.5%

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

   13. associate-/r/ 99.5%

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

   14. associate-/l/ 99.5%

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

   15. associate-/r/ 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 11: 61.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (sqrt (/ 0.1111111111111111 x)) (- (sqrt (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = -sqrt((x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = -sqrt((x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = -Math.sqrt((x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = -math.sqrt((x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(-sqrt(Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = -sqrt((x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], (-N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   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-/r* 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-/r* 99.4%

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

      4. inv-pow 99.4%

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

      5. *-commutative 99.4%

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

      6. unpow-prod-down 99.3%

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

      7. metadata-eval 99.3%

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

      8. inv-pow 99.3%

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

      9. div-inv 99.3%

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

      10. associate-+r- 99.3%

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

      11. add-sqr-sqrt 79.9%

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

      12. pow2 79.9%

          \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{\frac{0.1111111111111111}{x} + \left(y - 1\right)}\right)}^{2}} \]

   3. Applied egg-rr 79.9%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 79.6%

         \[\leadsto \color{blue}{\sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \cdot \sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}}} \]

      2. sqrt-unprod 74.1%

         \[\leadsto \color{blue}{\sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)}} \]

      3. unpow2 74.1%

         \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)} \cdot \sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      4. add-sqr-sqrt 74.3%

         \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      5. *-commutative 74.3%

         \[\leadsto \sqrt{\color{blue}{\left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      6. unpow2 74.3%

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

      7. add-sqr-sqrt 74.4%

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

      8. *-commutative 74.4%

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

      9. swap-sqr 40.5%

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

   5. Applied egg-rr 40.5%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{0.1111111111111111}{x} + \left(-1 + y\right)\right)}^{2} \cdot \left(x \cdot 9\right)}} \]

   6. Taylor expanded in x around 0 32.7%

      \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{{x}^{2}}} \cdot \left(x \cdot 9\right)} \]
   7. Step-by-step derivation

      1. unpow2 32.7%

         \[\leadsto \sqrt{\frac{0.012345679012345678}{\color{blue}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

   8. Simplified 32.7%

      \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

   9. Taylor expanded in x around 0 61.8%

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

   if 0.112000000000000002 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.5%

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

   3. Taylor expanded in y around 0 49.1%

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

      1. *-commutative 49.1%

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

   5. Simplified 49.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 2.0%

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

      3. pow2 2.0%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{x} \cdot -3\right)}^{2}}} \]

   7. Applied egg-rr 2.0%

      \[\leadsto \color{blue}{\sqrt{{\left(\sqrt{x} \cdot -3\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 2.0%

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

      2. swap-sqr 2.0%

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

      3. rem-square-sqrt 2.0%

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

      4. metadata-eval 2.0%

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

   9. Simplified 2.0%

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

       1. sqrt-prod 2.0%

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

       2. metadata-eval 2.0%

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

       3. metadata-eval 2.0%

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

       4. distribute-rgt-neg-in 2.0%

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

       5. add-cbrt-cube 2.0%

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

       6. metadata-eval 2.0%

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

       7. metadata-eval 2.0%

          \[\leadsto -\sqrt{x} \cdot \sqrt[3]{\color{blue}{-27}} \]

       8. add-sqr-sqrt 0.0%

          \[\leadsto -\color{blue}{\sqrt{\sqrt{x} \cdot \sqrt[3]{-27}} \cdot \sqrt{\sqrt{x} \cdot \sqrt[3]{-27}}} \]

       9. distribute-rgt-neg-in 0.0%

          \[\leadsto \color{blue}{\sqrt{\sqrt{x} \cdot \sqrt[3]{-27}} \cdot \left(-\sqrt{\sqrt{x} \cdot \sqrt[3]{-27}}\right)} \]

   11. Applied egg-rr 49.0%

       \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.25} \cdot \left(-{\left(x \cdot 9\right)}^{0.25}\right)} \]
   12. Step-by-step derivation

       1. distribute-rgt-neg-out 49.0%

          \[\leadsto \color{blue}{-{\left(x \cdot 9\right)}^{0.25} \cdot {\left(x \cdot 9\right)}^{0.25}} \]

       2. pow-sqr 49.1%

          \[\leadsto -\color{blue}{{\left(x \cdot 9\right)}^{\left(2 \cdot 0.25\right)}} \]

       3. metadata-eval 49.1%

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

       4. unpow1/2 49.1%

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

   13. Simplified 49.1%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \end{array} \]

Alternative 12: 61.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		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 <= 0.112d0) 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 <= 0.112) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		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 <= 0.112)
		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 <= 0.112)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], 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 < 0.112000000000000002

   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-/r* 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-/r* 99.4%

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

      4. inv-pow 99.4%

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

      5. *-commutative 99.4%

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

      6. unpow-prod-down 99.3%

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

      7. metadata-eval 99.3%

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

      8. inv-pow 99.3%

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

      9. div-inv 99.3%

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

      10. associate-+r- 99.3%

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

      11. add-sqr-sqrt 79.9%

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

      12. pow2 79.9%

          \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{\frac{0.1111111111111111}{x} + \left(y - 1\right)}\right)}^{2}} \]

   3. Applied egg-rr 79.9%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 79.6%

         \[\leadsto \color{blue}{\sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \cdot \sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}}} \]

      2. sqrt-unprod 74.1%

         \[\leadsto \color{blue}{\sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)}} \]

      3. unpow2 74.1%

         \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)} \cdot \sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      4. add-sqr-sqrt 74.3%

         \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      5. *-commutative 74.3%

         \[\leadsto \sqrt{\color{blue}{\left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

      6. unpow2 74.3%

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

      7. add-sqr-sqrt 74.4%

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

      8. *-commutative 74.4%

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

      9. swap-sqr 40.5%

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

   5. Applied egg-rr 40.5%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{0.1111111111111111}{x} + \left(-1 + y\right)\right)}^{2} \cdot \left(x \cdot 9\right)}} \]

   6. Taylor expanded in x around 0 32.7%

      \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{{x}^{2}}} \cdot \left(x \cdot 9\right)} \]
   7. Step-by-step derivation

      1. unpow2 32.7%

         \[\leadsto \sqrt{\frac{0.012345679012345678}{\color{blue}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

   8. Simplified 32.7%

      \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

   9. Taylor expanded in x around 0 61.8%

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

   if 0.112000000000000002 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.5%

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

   3. Taylor expanded in y around 0 49.1%

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

      1. *-commutative 49.1%

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

   5. Simplified 49.1%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 13: 3.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (* x 9.0)))

C

double code(double x, double y) {
	return sqrt((x * 9.0));
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.sqrt((x * 9.0))

Julia

function code(x, y)
	return sqrt(Float64(x * 9.0))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((x * 9.0));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in x around inf 66.6%

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

3. Taylor expanded in y around 0 25.4%

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

   1. *-commutative 25.4%

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

5. Simplified 25.4%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 3.4%

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

   3. pow2 3.4%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{x} \cdot -3\right)}^{2}}} \]

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{\sqrt{{\left(\sqrt{x} \cdot -3\right)}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 3.4%

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

   2. swap-sqr 3.4%

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

   3. rem-square-sqrt 3.4%

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

   4. metadata-eval 3.4%

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

9. Simplified 3.4%

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

10. Final simplification 3.4%

    \[\leadsto \sqrt{x \cdot 9} \]

Alternative 14: 37.4% 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 99.5%

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

   1. associate-/r* 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-/r* 99.5%

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

   4. inv-pow 99.5%

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

   5. *-commutative 99.5%

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

   6. unpow-prod-down 99.5%

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

   7. metadata-eval 99.5%

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

   8. inv-pow 99.5%

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

   9. div-inv 99.5%

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

   10. associate-+r- 99.5%

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

   11. add-sqr-sqrt 52.3%

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

   12. pow2 52.3%

       \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{\frac{0.1111111111111111}{x} + \left(y - 1\right)}\right)}^{2}} \]

3. Applied egg-rr 52.3%

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 52.1%

      \[\leadsto \color{blue}{\sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}} \cdot \sqrt{\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}}} \]

   2. sqrt-unprod 44.3%

      \[\leadsto \color{blue}{\sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)}} \]

   3. unpow2 44.3%

      \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)} \cdot \sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

   4. add-sqr-sqrt 44.4%

      \[\leadsto \sqrt{\left(\left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right) \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

   5. *-commutative 44.4%

      \[\leadsto \sqrt{\color{blue}{\left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)}^{2}\right)} \]

   6. unpow2 44.4%

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

   7. add-sqr-sqrt 44.8%

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

   8. *-commutative 44.8%

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

   9. swap-sqr 27.7%

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

5. Applied egg-rr 27.7%

   \[\leadsto \color{blue}{\sqrt{{\left(\frac{0.1111111111111111}{x} + \left(-1 + y\right)\right)}^{2} \cdot \left(x \cdot 9\right)}} \]

6. Taylor expanded in x around 0 17.5%

   \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{{x}^{2}}} \cdot \left(x \cdot 9\right)} \]
7. Step-by-step derivation

   1. unpow2 17.5%

      \[\leadsto \sqrt{\frac{0.012345679012345678}{\color{blue}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

8. Simplified 17.5%

   \[\leadsto \sqrt{\color{blue}{\frac{0.012345679012345678}{x \cdot x}} \cdot \left(x \cdot 9\right)} \]

9. Taylor expanded in x around 0 32.1%

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

10. Final simplification 32.1%

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

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 2023199 
(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)))