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

Percentage Accurate: 99.4% → 99.3%

Time: 10.2s

Alternatives: 11

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: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate--l+ 99.3%

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

   3. distribute-lft-in 99.3%

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

   4. +-commutative 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*r* 99.3%

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

   7. cancel-sign-sub 99.3%

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

   8. *-commutative 99.3%

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

   9. associate-*r* 99.4%

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

   10. *-commutative 99.4%

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

   11. distribute-rgt-out-- 99.4%

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

   12. distribute-lft-neg-in 99.4%

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

   13. cancel-sign-sub 99.4%

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

   14. +-commutative 99.4%

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

   15. *-commutative 99.4%

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

   16. distribute-rgt-in 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ t_1 := \sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+91}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-244}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0)) (t_1 (* (sqrt x) (/ 0.3333333333333333 x))))
   (if (<= y -8.5e+91)
     (* 3.0 (* (sqrt x) y))
     (if (<= y -9.5e-206)
       t_1
       (if (<= y -3e-244)
         t_0
         (if (<= y 9e-94)
           t_1
           (if (<= y 1e-8)
             t_0
             (if (<= y 1.52e+31) t_1 (* (sqrt x) (* 3.0 y))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double t_1 = sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (y <= -8.5e+91) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= -9.5e-206) {
		tmp = t_1;
	} else if (y <= -3e-244) {
		tmp = t_0;
	} else if (y <= 9e-94) {
		tmp = t_1;
	} else if (y <= 1e-8) {
		tmp = t_0;
	} else if (y <= 1.52e+31) {
		tmp = t_1;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    t_1 = sqrt(x) * (0.3333333333333333d0 / x)
    if (y <= (-8.5d+91)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= (-9.5d-206)) then
        tmp = t_1
    else if (y <= (-3d-244)) then
        tmp = t_0
    else if (y <= 9d-94) then
        tmp = t_1
    else if (y <= 1d-8) then
        tmp = t_0
    else if (y <= 1.52d+31) then
        tmp = t_1
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double t_1 = Math.sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (y <= -8.5e+91) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= -9.5e-206) {
		tmp = t_1;
	} else if (y <= -3e-244) {
		tmp = t_0;
	} else if (y <= 9e-94) {
		tmp = t_1;
	} else if (y <= 1e-8) {
		tmp = t_0;
	} else if (y <= 1.52e+31) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	t_1 = math.sqrt(x) * (0.3333333333333333 / x)
	tmp = 0
	if y <= -8.5e+91:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= -9.5e-206:
		tmp = t_1
	elif y <= -3e-244:
		tmp = t_0
	elif y <= 9e-94:
		tmp = t_1
	elif y <= 1e-8:
		tmp = t_0
	elif y <= 1.52e+31:
		tmp = t_1
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	t_1 = Float64(sqrt(x) * Float64(0.3333333333333333 / x))
	tmp = 0.0
	if (y <= -8.5e+91)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= -9.5e-206)
		tmp = t_1;
	elseif (y <= -3e-244)
		tmp = t_0;
	elseif (y <= 9e-94)
		tmp = t_1;
	elseif (y <= 1e-8)
		tmp = t_0;
	elseif (y <= 1.52e+31)
		tmp = t_1;
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	t_1 = sqrt(x) * (0.3333333333333333 / x);
	tmp = 0.0;
	if (y <= -8.5e+91)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= -9.5e-206)
		tmp = t_1;
	elseif (y <= -3e-244)
		tmp = t_0;
	elseif (y <= 9e-94)
		tmp = t_1;
	elseif (y <= 1e-8)
		tmp = t_0;
	elseif (y <= 1.52e+31)
		tmp = t_1;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.5e+91], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -9.5e-206], t$95$1, If[LessEqual[y, -3e-244], t$95$0, If[LessEqual[y, 9e-94], t$95$1, If[LessEqual[y, 1e-8], t$95$0, If[LessEqual[y, 1.52e+31], t$95$1, N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.4999999999999995e91

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 79.4%

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

   if -8.4999999999999995e91 < y < -9.49999999999999979e-206 or -3.0000000000000001e-244 < y < 9.0000000000000004e-94 or 1e-8 < y < 1.5200000000000001e31

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. associate--l+ 99.2%

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

      3. distribute-lft-in 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-*r* 99.2%

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

      7. cancel-sign-sub 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-*r* 99.2%

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

      10. *-commutative 99.2%

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

      11. distribute-rgt-out-- 99.3%

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

      12. distribute-lft-neg-in 99.3%

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

      13. cancel-sign-sub 99.3%

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

      14. +-commutative 99.3%

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

      15. *-commutative 99.3%

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

      16. distribute-rgt-in 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*r/ 93.7%

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

      3. metadata-eval 93.7%

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

      4. sub-neg 93.7%

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

      5. metadata-eval 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in x around 0 62.5%

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

   if -9.49999999999999979e-206 < y < -3.0000000000000001e-244 or 9.0000000000000004e-94 < y < 1e-8

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.5%

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

      7. cancel-sign-sub 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out-- 99.6%

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

      12. distribute-lft-neg-in 99.6%

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

      13. cancel-sign-sub 99.6%

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

      14. +-commutative 99.6%

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

      15. *-commutative 99.6%

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

      16. distribute-rgt-in 99.6%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 73.8%

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

   5. Taylor expanded in y around 0 71.6%

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

      1. *-commutative 71.6%

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

   7. Simplified 71.6%

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

   if 1.5200000000000001e31 < y

   1. Initial program 99.5%

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

      1. +-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-lft-in 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.6%

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

      7. cancel-sign-sub 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out-- 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. cancel-sign-sub 99.5%

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

      14. +-commutative 99.5%

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

      15. *-commutative 99.5%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 87.5%

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

      1. associate-*r* 87.5%

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

      2. *-commutative 87.5%

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

   6. Simplified 87.5%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+91}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;y \leq 10^{-8}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]

Alternative 3: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ t_1 := \sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+92}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0)) (t_1 (* (sqrt x) (/ 0.3333333333333333 x))))
   (if (<= y -1.05e+92)
     (* 3.0 (* (sqrt x) y))
     (if (<= y -8.8e-206)
       t_1
       (if (<= y -1.9e-242)
         t_0
         (if (<= y 9e-94)
           t_1
           (if (<= y 1e-8)
             t_0
             (if (<= y 5.6e+31) t_1 (* y (sqrt (* x 9.0)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double t_1 = sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (y <= -1.05e+92) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= -8.8e-206) {
		tmp = t_1;
	} else if (y <= -1.9e-242) {
		tmp = t_0;
	} else if (y <= 9e-94) {
		tmp = t_1;
	} else if (y <= 1e-8) {
		tmp = t_0;
	} else if (y <= 5.6e+31) {
		tmp = t_1;
	} else {
		tmp = y * 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    t_1 = sqrt(x) * (0.3333333333333333d0 / x)
    if (y <= (-1.05d+92)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= (-8.8d-206)) then
        tmp = t_1
    else if (y <= (-1.9d-242)) then
        tmp = t_0
    else if (y <= 9d-94) then
        tmp = t_1
    else if (y <= 1d-8) then
        tmp = t_0
    else if (y <= 5.6d+31) then
        tmp = t_1
    else
        tmp = y * sqrt((x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double t_1 = Math.sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (y <= -1.05e+92) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= -8.8e-206) {
		tmp = t_1;
	} else if (y <= -1.9e-242) {
		tmp = t_0;
	} else if (y <= 9e-94) {
		tmp = t_1;
	} else if (y <= 1e-8) {
		tmp = t_0;
	} else if (y <= 5.6e+31) {
		tmp = t_1;
	} else {
		tmp = y * Math.sqrt((x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	t_1 = math.sqrt(x) * (0.3333333333333333 / x)
	tmp = 0
	if y <= -1.05e+92:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= -8.8e-206:
		tmp = t_1
	elif y <= -1.9e-242:
		tmp = t_0
	elif y <= 9e-94:
		tmp = t_1
	elif y <= 1e-8:
		tmp = t_0
	elif y <= 5.6e+31:
		tmp = t_1
	else:
		tmp = y * math.sqrt((x * 9.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	t_1 = Float64(sqrt(x) * Float64(0.3333333333333333 / x))
	tmp = 0.0
	if (y <= -1.05e+92)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= -8.8e-206)
		tmp = t_1;
	elseif (y <= -1.9e-242)
		tmp = t_0;
	elseif (y <= 9e-94)
		tmp = t_1;
	elseif (y <= 1e-8)
		tmp = t_0;
	elseif (y <= 5.6e+31)
		tmp = t_1;
	else
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	t_1 = sqrt(x) * (0.3333333333333333 / x);
	tmp = 0.0;
	if (y <= -1.05e+92)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= -8.8e-206)
		tmp = t_1;
	elseif (y <= -1.9e-242)
		tmp = t_0;
	elseif (y <= 9e-94)
		tmp = t_1;
	elseif (y <= 1e-8)
		tmp = t_0;
	elseif (y <= 5.6e+31)
		tmp = t_1;
	else
		tmp = y * sqrt((x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+92], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -8.8e-206], t$95$1, If[LessEqual[y, -1.9e-242], t$95$0, If[LessEqual[y, 9e-94], t$95$1, If[LessEqual[y, 1e-8], t$95$0, If[LessEqual[y, 5.6e+31], t$95$1, N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.04999999999999993e92

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 79.4%

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

   if -1.04999999999999993e92 < y < -8.7999999999999995e-206 or -1.9000000000000001e-242 < y < 9.0000000000000004e-94 or 1e-8 < y < 5.60000000000000034e31

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. associate--l+ 99.2%

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

      3. distribute-lft-in 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-*r* 99.2%

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

      7. cancel-sign-sub 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-*r* 99.2%

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

      10. *-commutative 99.2%

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

      11. distribute-rgt-out-- 99.3%

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

      12. distribute-lft-neg-in 99.3%

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

      13. cancel-sign-sub 99.3%

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

      14. +-commutative 99.3%

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

      15. *-commutative 99.3%

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

      16. distribute-rgt-in 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*r/ 93.7%

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

      3. metadata-eval 93.7%

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

      4. sub-neg 93.7%

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

      5. metadata-eval 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in x around 0 62.5%

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

   if -8.7999999999999995e-206 < y < -1.9000000000000001e-242 or 9.0000000000000004e-94 < y < 1e-8

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.5%

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

      7. cancel-sign-sub 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out-- 99.6%

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

      12. distribute-lft-neg-in 99.6%

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

      13. cancel-sign-sub 99.6%

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

      14. +-commutative 99.6%

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

      15. *-commutative 99.6%

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

      16. distribute-rgt-in 99.6%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 73.8%

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

   5. Taylor expanded in y around 0 71.6%

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

      1. *-commutative 71.6%

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

   7. Simplified 71.6%

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

   if 5.60000000000000034e31 < y

   1. Initial program 99.5%

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

      1. +-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 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.5%

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

      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. Step-by-step derivation

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 87.7%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+92}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;y \leq 10^{-8}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \end{array} \]

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+91}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+31}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e+91)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 3.1e+31)
     (* 3.0 (* (sqrt x) (+ (/ 0.1111111111111111 x) -1.0)))
     (* y (sqrt (* x 9.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e+91) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 3.1e+31) {
		tmp = 3.0 * (sqrt(x) * ((0.1111111111111111 / x) + -1.0));
	} else {
		tmp = y * 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 (y <= (-9d+91)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 3.1d+31) then
        tmp = 3.0d0 * (sqrt(x) * ((0.1111111111111111d0 / x) + (-1.0d0)))
    else
        tmp = y * sqrt((x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9e+91) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 3.1e+31) {
		tmp = 3.0 * (Math.sqrt(x) * ((0.1111111111111111 / x) + -1.0));
	} else {
		tmp = y * Math.sqrt((x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e+91:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 3.1e+31:
		tmp = 3.0 * (math.sqrt(x) * ((0.1111111111111111 / x) + -1.0))
	else:
		tmp = y * math.sqrt((x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e+91)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 3.1e+31)
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(Float64(0.1111111111111111 / x) + -1.0)));
	else
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9e+91)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 3.1e+31)
		tmp = 3.0 * (sqrt(x) * ((0.1111111111111111 / x) + -1.0));
	else
		tmp = y * sqrt((x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e+91], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+31], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9e91

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 79.4%

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

   if -9e91 < y < 3.1000000000000002e31

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. associate-/r* 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around 0 94.2%

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

   5. Taylor expanded in x around 0 94.2%

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

   if 3.1000000000000002e31 < y

   1. Initial program 99.5%

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

      1. +-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 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.5%

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

      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. Step-by-step derivation

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 87.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+91}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+31}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (sqrt(x) * 3.0) * ((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 = (sqrt(x) * 3.0d0) * ((y + (1.0d0 / (x * 9.0d0))) + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Final simplification 99.3%

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

Alternative 6: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+91}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.8e+91)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 7.8e+30)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (* y (sqrt (* x 9.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.8e+91) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 7.8e+30) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * 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 (y <= (-9.8d+91)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 7.8d+30) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = y * sqrt((x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.8e+91) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 7.8e+30) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * Math.sqrt((x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.8e+91:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 7.8e+30:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = y * math.sqrt((x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.8e+91)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 7.8e+30)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.8e+91)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 7.8e+30)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = y * sqrt((x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.8e+91], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e+30], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.8000000000000006e91

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 79.4%

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

   if -9.8000000000000006e91 < y < 7.80000000000000021e30

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-lft-in 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.2%

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

      7. cancel-sign-sub 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-*r* 99.3%

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

      10. *-commutative 99.3%

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

      11. distribute-rgt-out-- 99.3%

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

      12. distribute-lft-neg-in 99.3%

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

      13. cancel-sign-sub 99.3%

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

      14. +-commutative 99.3%

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

      15. *-commutative 99.3%

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

      16. distribute-rgt-in 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 94.2%

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

      1. *-commutative 94.2%

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

      2. associate-*r/ 94.2%

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

      3. metadata-eval 94.2%

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

      4. sub-neg 94.2%

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

      5. metadata-eval 94.2%

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

   6. Simplified 94.2%

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

   if 7.80000000000000021e30 < y

   1. Initial program 99.5%

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

      1. +-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 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.5%

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

      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. Step-by-step derivation

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 87.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+91}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \end{array} \]

Alternative 7: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate--l+ 99.3%

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

   3. distribute-rgt-in 99.3%

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

   4. remove-double-neg 99.3%

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

   5. distribute-lft-neg-in 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. mul-1-neg 99.3%

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

   8. metadata-eval 99.3%

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

   9. *-commutative 99.3%

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

   10. associate-/r* 99.3%

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

   11. distribute-neg-frac 99.3%

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

   12. *-commutative 99.3%

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

   13. associate-/r/ 99.3%

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

   14. associate-/l/ 99.3%

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

   15. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. metadata-eval 99.3%

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

   3. sqrt-prod 98.7%

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

   4. pow1/2 98.7%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

5. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
6. Step-by-step derivation

   1. unpow1/2 98.7%

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

7. Simplified 98.7%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
8. Step-by-step derivation

   1. sqrt-prod 99.3%

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

   2. add-sqr-sqrt 99.0%

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

   3. metadata-eval 99.0%

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

   4. associate-*l* 99.1%

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

   5. pow1/2 99.1%

      \[\leadsto \left(\sqrt{\color{blue}{{x}^{0.5}}} \cdot \left(\sqrt{\sqrt{x}} \cdot 3\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. sqrt-pow1 99.2%

      \[\leadsto \left(\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{x}} \cdot 3\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   7. metadata-eval 99.2%

      \[\leadsto \left({x}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{x}} \cdot 3\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   8. pow1/2 99.2%

      \[\leadsto \left({x}^{0.25} \cdot \left(\sqrt{\color{blue}{{x}^{0.5}}} \cdot 3\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   9. sqrt-pow1 99.1%

      \[\leadsto \left({x}^{0.25} \cdot \left(\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}} \cdot 3\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   10. metadata-eval 99.1%

       \[\leadsto \left({x}^{0.25} \cdot \left({x}^{\color{blue}{0.25}} \cdot 3\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

9. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\left({x}^{0.25} \cdot \left({x}^{0.25} \cdot 3\right)\right)} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

10. Taylor expanded in y around 0 99.3%

    \[\leadsto \color{blue}{3 \cdot \left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \sqrt{x}\right) + 3 \cdot \left(y \cdot \sqrt{x}\right)} \]
11. 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. metadata-eval 99.3%

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

    5. associate-*r/ 99.4%

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

    6. metadata-eval 99.4%

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

    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-out 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. +-commutative 99.3%

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

    11. +-commutative 99.3%

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

    12. +-commutative 99.3%

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

    13. associate-+l+ 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 8: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.3 \cdot 10^{-6}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 2.3e-6)))
   (* 3.0 (* (sqrt x) y))
   (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 2.3e-6)) {
		tmp = 3.0 * (sqrt(x) * y);
	} 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 ((y <= (-1.0d0)) .or. (.not. (y <= 2.3d-6))) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 2.3e-6)) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 2.3e-6):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 2.3e-6))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 2.3e-6)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 2.3e-6]], $MachinePrecision]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.3e-6 < y

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.5%

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

      7. cancel-sign-sub 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out-- 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. cancel-sign-sub 99.5%

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

      14. +-commutative 99.5%

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

      15. *-commutative 99.5%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 74.4%

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

   if -1 < y < 2.3e-6

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-lft-in 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.2%

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

      7. cancel-sign-sub 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-*r* 99.3%

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

      10. *-commutative 99.3%

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

      11. distribute-rgt-out-- 99.3%

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

      12. distribute-lft-neg-in 99.3%

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

      13. cancel-sign-sub 99.3%

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

      14. +-commutative 99.3%

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

      15. *-commutative 99.3%

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

      16. distribute-rgt-in 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around inf 44.0%

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

   5. Taylor expanded in y around 0 41.7%

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

      1. *-commutative 41.7%

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

   7. Simplified 41.7%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.3 \cdot 10^{-6}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 9: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 2.3e-6) (* (sqrt x) -3.0) (* (sqrt x) (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 2.3e-6) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 2.3d-6) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 2.3e-6) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 2.3e-6:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 2.3e-6)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 2.3e-6)
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e-6], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.3%

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

      7. cancel-sign-sub 99.3%

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

      8. *-commutative 99.3%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 64.1%

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

   if -1 < y < 2.3e-6

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-lft-in 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.2%

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

      7. cancel-sign-sub 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-*r* 99.3%

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

      10. *-commutative 99.3%

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

      11. distribute-rgt-out-- 99.3%

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

      12. distribute-lft-neg-in 99.3%

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

      13. cancel-sign-sub 99.3%

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

      14. +-commutative 99.3%

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

      15. *-commutative 99.3%

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

      16. distribute-rgt-in 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around inf 44.0%

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

   5. Taylor expanded in y around 0 41.7%

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

      1. *-commutative 41.7%

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

   7. Simplified 41.7%

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

   if 2.3e-6 < y

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.6%

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

      7. cancel-sign-sub 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out-- 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. cancel-sign-sub 99.5%

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

      14. +-commutative 99.5%

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

      15. *-commutative 99.5%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 84.8%

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

      1. associate-*r* 84.8%

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

      2. *-commutative 84.8%

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

   6. Simplified 84.8%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]

Alternative 10: 26.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11) (sqrt (* x 9.0)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = sqrt((x * 9.0));
	} 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 <= 0.11d0) then
        tmp = sqrt((x * 9.0d0))
    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 <= 0.11) {
		tmp = Math.sqrt((x * 9.0));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = math.sqrt((x * 9.0))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = sqrt(Float64(x * 9.0));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = sqrt((x * 9.0));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. associate--l+ 99.2%

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

      3. distribute-lft-in 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.3%

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

      7. cancel-sign-sub 99.3%

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

      8. *-commutative 99.3%

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

      9. associate-*r* 99.3%

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

      10. *-commutative 99.3%

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

      11. distribute-rgt-out-- 99.3%

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

      12. distribute-lft-neg-in 99.3%

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

      13. cancel-sign-sub 99.3%

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

      14. +-commutative 99.3%

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

      15. *-commutative 99.3%

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

      16. distribute-rgt-in 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around inf 24.7%

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

   5. Taylor expanded in y around 0 1.8%

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

      1. *-commutative 1.8%

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

   7. Simplified 1.8%

      \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
   8. 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 4.7%

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

      3. swap-sqr 4.7%

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

      4. add-sqr-sqrt 4.7%

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

      5. metadata-eval 4.7%

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

      6. add-sqr-sqrt 4.7%

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

      7. pow2 4.7%

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

      8. pow1/2 4.7%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}}\right)}^{2} \]

      9. sqrt-pow1 4.7%

         \[\leadsto {\color{blue}{\left({\left(x \cdot 9\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      10. metadata-eval 4.7%

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

   9. Applied egg-rr 4.7%

      \[\leadsto \color{blue}{{\left({\left(x \cdot 9\right)}^{0.25}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 4.7%

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

       2. pow-sqr 4.7%

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

       3. metadata-eval 4.7%

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

       4. unpow1/2 4.7%

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

   11. Simplified 4.7%

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

   if 0.110000000000000001 < x

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      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-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.5%

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

      7. cancel-sign-sub 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. cancel-sign-sub 99.5%

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

      14. +-commutative 99.5%

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

      15. *-commutative 99.5%

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

      16. distribute-rgt-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 99.1%

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

   5. Taylor expanded in y around 0 48.5%

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

      1. *-commutative 48.5%

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

   7. Simplified 48.5%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 11: 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 99.3%

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

   1. +-commutative 99.3%

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

   2. associate--l+ 99.3%

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

   3. distribute-lft-in 99.3%

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

   4. +-commutative 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*r* 99.3%

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

   7. cancel-sign-sub 99.3%

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

   8. *-commutative 99.3%

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

   9. associate-*r* 99.4%

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

   10. *-commutative 99.4%

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

   11. distribute-rgt-out-- 99.4%

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

   12. distribute-lft-neg-in 99.4%

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

   13. cancel-sign-sub 99.4%

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

   14. +-commutative 99.4%

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

   15. *-commutative 99.4%

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

   16. distribute-rgt-in 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around inf 58.7%

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

5. Taylor expanded in y around 0 23.2%

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

   1. *-commutative 23.2%

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

7. Simplified 23.2%

   \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
8. 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 3.6%

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

   3. swap-sqr 3.6%

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

   4. add-sqr-sqrt 3.6%

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

   5. metadata-eval 3.6%

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

   6. add-sqr-sqrt 3.6%

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

   7. pow2 3.6%

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

   8. pow1/2 3.6%

      \[\leadsto {\left(\sqrt{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}}\right)}^{2} \]

   9. sqrt-pow1 3.6%

      \[\leadsto {\color{blue}{\left({\left(x \cdot 9\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

   10. metadata-eval 3.6%

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

9. Applied egg-rr 3.6%

   \[\leadsto \color{blue}{{\left({\left(x \cdot 9\right)}^{0.25}\right)}^{2}} \]
10. Step-by-step derivation

    1. unpow2 3.6%

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

    2. pow-sqr 3.6%

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

    3. metadata-eval 3.6%

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

    4. unpow1/2 3.6%

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

11. Simplified 3.6%

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

12. Final simplification 3.6%

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

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 2023174 
(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)))