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

Percentage Accurate: 99.4% → 99.4%

Time: 11.2s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-def 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.3%

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

   13. associate-*r/ 99.4%

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

   14. metadata-eval 99.4%

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

   15. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. clear-num 99.3%

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

   2. inv-pow 99.3%

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

   3. div-inv 99.4%

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

   4. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

   \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3 + \color{blue}{{\left(x \cdot 3\right)}^{-1}}\right) \]
6. Step-by-step derivation

   1. unpow-1 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-def 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.3%

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

   13. associate-*r/ 99.4%

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

   14. metadata-eval 99.4%

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

   15. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 3: 59.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_1 := \sqrt{x} \cdot -3\\ t_2 := \frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.7 \cdot 10^{+128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+125}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (* (sqrt x) y)))
        (t_1 (* (sqrt x) -3.0))
        (t_2 (/ (sqrt x) (* x 3.0))))
   (if (<= y -5.4e+146)
     t_0
     (if (<= y -6.7e+128)
       t_2
       (if (<= y -1.0)
         (* y (* (sqrt x) 3.0))
         (if (<= y 3.75e-196)
           t_1
           (if (<= y 1.5e-144)
             t_2
             (if (<= y 3e-25) t_1 (if (<= y 3.6e+125) t_2 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (sqrt(x) * y);
	double t_1 = sqrt(x) * -3.0;
	double t_2 = sqrt(x) / (x * 3.0);
	double tmp;
	if (y <= -5.4e+146) {
		tmp = t_0;
	} else if (y <= -6.7e+128) {
		tmp = t_2;
	} else if (y <= -1.0) {
		tmp = y * (sqrt(x) * 3.0);
	} else if (y <= 3.75e-196) {
		tmp = t_1;
	} else if (y <= 1.5e-144) {
		tmp = t_2;
	} else if (y <= 3e-25) {
		tmp = t_1;
	} else if (y <= 3.6e+125) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 3.0d0 * (sqrt(x) * y)
    t_1 = sqrt(x) * (-3.0d0)
    t_2 = sqrt(x) / (x * 3.0d0)
    if (y <= (-5.4d+146)) then
        tmp = t_0
    else if (y <= (-6.7d+128)) then
        tmp = t_2
    else if (y <= (-1.0d0)) then
        tmp = y * (sqrt(x) * 3.0d0)
    else if (y <= 3.75d-196) then
        tmp = t_1
    else if (y <= 1.5d-144) then
        tmp = t_2
    else if (y <= 3d-25) then
        tmp = t_1
    else if (y <= 3.6d+125) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * (Math.sqrt(x) * y);
	double t_1 = Math.sqrt(x) * -3.0;
	double t_2 = Math.sqrt(x) / (x * 3.0);
	double tmp;
	if (y <= -5.4e+146) {
		tmp = t_0;
	} else if (y <= -6.7e+128) {
		tmp = t_2;
	} else if (y <= -1.0) {
		tmp = y * (Math.sqrt(x) * 3.0);
	} else if (y <= 3.75e-196) {
		tmp = t_1;
	} else if (y <= 1.5e-144) {
		tmp = t_2;
	} else if (y <= 3e-25) {
		tmp = t_1;
	} else if (y <= 3.6e+125) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * (math.sqrt(x) * y)
	t_1 = math.sqrt(x) * -3.0
	t_2 = math.sqrt(x) / (x * 3.0)
	tmp = 0
	if y <= -5.4e+146:
		tmp = t_0
	elif y <= -6.7e+128:
		tmp = t_2
	elif y <= -1.0:
		tmp = y * (math.sqrt(x) * 3.0)
	elif y <= 3.75e-196:
		tmp = t_1
	elif y <= 1.5e-144:
		tmp = t_2
	elif y <= 3e-25:
		tmp = t_1
	elif y <= 3.6e+125:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(sqrt(x) * y))
	t_1 = Float64(sqrt(x) * -3.0)
	t_2 = Float64(sqrt(x) / Float64(x * 3.0))
	tmp = 0.0
	if (y <= -5.4e+146)
		tmp = t_0;
	elseif (y <= -6.7e+128)
		tmp = t_2;
	elseif (y <= -1.0)
		tmp = Float64(y * Float64(sqrt(x) * 3.0));
	elseif (y <= 3.75e-196)
		tmp = t_1;
	elseif (y <= 1.5e-144)
		tmp = t_2;
	elseif (y <= 3e-25)
		tmp = t_1;
	elseif (y <= 3.6e+125)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * (sqrt(x) * y);
	t_1 = sqrt(x) * -3.0;
	t_2 = sqrt(x) / (x * 3.0);
	tmp = 0.0;
	if (y <= -5.4e+146)
		tmp = t_0;
	elseif (y <= -6.7e+128)
		tmp = t_2;
	elseif (y <= -1.0)
		tmp = y * (sqrt(x) * 3.0);
	elseif (y <= 3.75e-196)
		tmp = t_1;
	elseif (y <= 1.5e-144)
		tmp = t_2;
	elseif (y <= 3e-25)
		tmp = t_1;
	elseif (y <= 3.6e+125)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.4e+146], t$95$0, If[LessEqual[y, -6.7e+128], t$95$2, If[LessEqual[y, -1.0], N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.75e-196], t$95$1, If[LessEqual[y, 1.5e-144], t$95$2, If[LessEqual[y, 3e-25], t$95$1, If[LessEqual[y, 3.6e+125], t$95$2, t$95$0]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.39999999999999977e146 or 3.6000000000000003e125 < 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%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.4%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 88.0%

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

   if -5.39999999999999977e146 < y < -6.69999999999999993e128 or 3.75e-196 < y < 1.4999999999999999e-144 or 2.9999999999999998e-25 < y < 3.6000000000000003e125

   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. metadata-eval 99.4%

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

      2. sqrt-prod 99.6%

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

      3. *-commutative 99.6%

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

      4. pow1/2 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 69.3%

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

      1. associate-*r/ 69.3%

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

      2. metadata-eval 69.3%

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

      3. sub-neg 69.3%

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

      4. metadata-eval 69.3%

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

      5. associate-*l* 69.3%

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

      6. *-commutative 69.3%

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

      7. associate-*l* 69.2%

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

      8. +-commutative 69.2%

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

      9. distribute-rgt-in 69.2%

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

      10. metadata-eval 69.2%

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

   8. Simplified 69.2%

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

      1. flip-+ 29.8%

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

      2. associate-*r/ 29.8%

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

      3. associate-/l* 29.8%

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

      4. *-un-lft-identity 29.8%

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

      5. associate-/l* 29.8%

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

      6. flip-+ 69.2%

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

      7. associate-*l/ 69.2%

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

      8. metadata-eval 69.2%

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

   10. Applied egg-rr 69.2%

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

   11. Taylor expanded in x around 0 63.9%

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

   if -6.69999999999999993e128 < 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%

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

      2. associate-*l* 99.1%

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

      3. associate--l+ 99.1%

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

      4. distribute-lft-in 99.1%

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

      5. fma-def 99.2%

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

      6. sub-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. distribute-lft-in 99.2%

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

      9. metadata-eval 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

      3. div-inv 99.3%

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

      4. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in y around inf 50.7%

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

      1. associate-*r* 50.9%

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

      2. *-commutative 50.9%

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

   8. Simplified 50.9%

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

   if -1 < y < 3.75e-196 or 1.4999999999999999e-144 < y < 2.9999999999999998e-25

   1. Initial program 99.3%

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

   2. Taylor expanded in y around inf 63.4%

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

   3. Taylor expanded in y around 0 61.5%

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

      1. *-commutative 61.5%

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

   5. Simplified 61.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+146}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -6.7 \cdot 10^{+128}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 4: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-55}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{3}{x \cdot 9}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.5e-83)
   (/ (sqrt x) (* x 3.0))
   (if (<= x 4.4e-55)
     (* 3.0 (* (sqrt x) y))
     (if (<= x 8.5e-15)
       (* (sqrt x) (+ -3.0 (/ 3.0 (* x 9.0))))
       (* 3.0 (* (sqrt x) (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.5e-83) {
		tmp = sqrt(x) / (x * 3.0);
	} else if (x <= 4.4e-55) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (x <= 8.5e-15) {
		tmp = sqrt(x) * (-3.0 + (3.0 / (x * 9.0)));
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6.5d-83) then
        tmp = sqrt(x) / (x * 3.0d0)
    else if (x <= 4.4d-55) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (x <= 8.5d-15) then
        tmp = sqrt(x) * ((-3.0d0) + (3.0d0 / (x * 9.0d0)))
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.5e-83) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else if (x <= 4.4e-55) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (x <= 8.5e-15) {
		tmp = Math.sqrt(x) * (-3.0 + (3.0 / (x * 9.0)));
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.5e-83:
		tmp = math.sqrt(x) / (x * 3.0)
	elif x <= 4.4e-55:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif x <= 8.5e-15:
		tmp = math.sqrt(x) * (-3.0 + (3.0 / (x * 9.0)))
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.5e-83)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	elseif (x <= 4.4e-55)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (x <= 8.5e-15)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(3.0 / Float64(x * 9.0))));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.5e-83)
		tmp = sqrt(x) / (x * 3.0);
	elseif (x <= 4.4e-55)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (x <= 8.5e-15)
		tmp = sqrt(x) * (-3.0 + (3.0 / (x * 9.0)));
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.5e-83], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.4e-55], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-15], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(3.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.5e-83

   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. metadata-eval 99.2%

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

      2. sqrt-prod 99.4%

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

      3. *-commutative 99.4%

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

      4. pow1/2 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. unpow1/2 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. metadata-eval 74.1%

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

      3. sub-neg 74.1%

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

      4. metadata-eval 74.1%

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

      5. associate-*l* 74.1%

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

      6. *-commutative 74.1%

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

      7. associate-*l* 74.1%

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

      8. +-commutative 74.1%

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

      9. distribute-rgt-in 74.1%

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

      10. metadata-eval 74.1%

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

   8. Simplified 74.1%

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

      1. flip-+ 19.1%

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

      2. associate-*r/ 19.1%

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

      3. associate-/l* 19.1%

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

      4. *-un-lft-identity 19.1%

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

      5. associate-/l* 19.1%

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

      6. flip-+ 74.1%

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

      7. associate-*l/ 74.2%

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

      8. metadata-eval 74.2%

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

   10. Applied egg-rr 74.2%

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

   11. Taylor expanded in x around 0 74.4%

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

   if 6.5e-83 < x < 4.3999999999999999e-55

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   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 inf 89.7%

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

   if 4.3999999999999999e-55 < x < 8.50000000000000007e-15

   1. Initial program 99.0%

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

      1. metadata-eval 99.0%

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

      2. sqrt-prod 99.5%

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

      3. *-commutative 99.5%

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

      4. pow1/2 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around 0 62.4%

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

      1. associate-*r/ 62.6%

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

      2. metadata-eval 62.6%

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

      3. sub-neg 62.6%

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

      4. metadata-eval 62.6%

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

      5. associate-*l* 62.5%

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

      6. *-commutative 62.5%

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

      7. associate-*l* 62.6%

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

      8. +-commutative 62.6%

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

      9. distribute-rgt-in 62.6%

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

      10. metadata-eval 62.6%

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

   8. Simplified 62.6%

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

      1. clear-num 62.2%

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

      2. associate-*l/ 62.3%

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

      3. metadata-eval 62.3%

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

      4. div-inv 62.7%

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

      5. metadata-eval 62.7%

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

   10. Applied egg-rr 62.7%

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

   if 8.50000000000000007e-15 < 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. Taylor expanded in y around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. flip-- 81.0%

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

      3. associate-*l/ 78.1%

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

      4. metadata-eval 78.1%

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

      5. sub-neg 78.1%

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

      6. pow2 78.1%

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

      7. metadata-eval 78.1%

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

   4. Applied egg-rr 78.1%

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

      1. unpow2 78.1%

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

      2. difference-of-sqr--1 78.1%

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

      3. difference-of-sqr-1 78.1%

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

      4. metadata-eval 78.1%

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

      5. associate-*l/ 81.0%

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

      6. flip-- 98.3%

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

      7. *-commutative 98.3%

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

      8. associate-*r* 98.3%

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

      9. sub-neg 98.3%

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

      10. metadata-eval 98.3%

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

   6. Applied egg-rr 98.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-55}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{3}{x \cdot 9}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 5: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-55}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{\frac{1}{x}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.5e-83)
   (/ (sqrt x) (* x 3.0))
   (if (<= x 5e-55)
     (* 3.0 (* (sqrt x) y))
     (if (<= x 9.2e-15)
       (* (sqrt x) (+ -3.0 (/ (/ 1.0 x) 3.0)))
       (* 3.0 (* (sqrt x) (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.5e-83) {
		tmp = sqrt(x) / (x * 3.0);
	} else if (x <= 5e-55) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (x <= 9.2e-15) {
		tmp = sqrt(x) * (-3.0 + ((1.0 / x) / 3.0));
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 4.5d-83) then
        tmp = sqrt(x) / (x * 3.0d0)
    else if (x <= 5d-55) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (x <= 9.2d-15) then
        tmp = sqrt(x) * ((-3.0d0) + ((1.0d0 / x) / 3.0d0))
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4.5e-83) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else if (x <= 5e-55) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (x <= 9.2e-15) {
		tmp = Math.sqrt(x) * (-3.0 + ((1.0 / x) / 3.0));
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.5e-83:
		tmp = math.sqrt(x) / (x * 3.0)
	elif x <= 5e-55:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif x <= 9.2e-15:
		tmp = math.sqrt(x) * (-3.0 + ((1.0 / x) / 3.0))
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.5e-83)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	elseif (x <= 5e-55)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (x <= 9.2e-15)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(Float64(1.0 / x) / 3.0)));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.5e-83)
		tmp = sqrt(x) / (x * 3.0);
	elseif (x <= 5e-55)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (x <= 9.2e-15)
		tmp = sqrt(x) * (-3.0 + ((1.0 / x) / 3.0));
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.5e-83], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-55], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.2e-15], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(N[(1.0 / x), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.49999999999999997e-83

   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. metadata-eval 99.2%

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

      2. sqrt-prod 99.4%

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

      3. *-commutative 99.4%

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

      4. pow1/2 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. unpow1/2 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. metadata-eval 74.1%

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

      3. sub-neg 74.1%

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

      4. metadata-eval 74.1%

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

      5. associate-*l* 74.1%

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

      6. *-commutative 74.1%

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

      7. associate-*l* 74.1%

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

      8. +-commutative 74.1%

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

      9. distribute-rgt-in 74.1%

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

      10. metadata-eval 74.1%

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

   8. Simplified 74.1%

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

      1. flip-+ 19.1%

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

      2. associate-*r/ 19.1%

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

      3. associate-/l* 19.1%

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

      4. *-un-lft-identity 19.1%

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

      5. associate-/l* 19.1%

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

      6. flip-+ 74.1%

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

      7. associate-*l/ 74.2%

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

      8. metadata-eval 74.2%

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

   10. Applied egg-rr 74.2%

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

   11. Taylor expanded in x around 0 74.4%

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

   if 4.49999999999999997e-83 < x < 5.0000000000000002e-55

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   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 inf 89.7%

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

   if 5.0000000000000002e-55 < x < 9.19999999999999961e-15

   1. Initial program 99.0%

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

      1. metadata-eval 99.0%

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

      2. sqrt-prod 99.5%

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

      3. *-commutative 99.5%

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

      4. pow1/2 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around 0 62.4%

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

      1. associate-*r/ 62.6%

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

      2. metadata-eval 62.6%

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

      3. sub-neg 62.6%

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

      4. metadata-eval 62.6%

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

      5. associate-*l* 62.5%

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

      6. *-commutative 62.5%

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

      7. associate-*l* 62.6%

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

      8. +-commutative 62.6%

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

      9. distribute-rgt-in 62.6%

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

      10. metadata-eval 62.6%

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

   8. Simplified 62.6%

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

      1. associate-*l/ 62.6%

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

      2. metadata-eval 62.6%

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

      3. metadata-eval 62.6%

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

      4. associate-/r* 62.8%

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

      5. associate-/l/ 62.8%

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

   10. Applied egg-rr 62.8%

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

   if 9.19999999999999961e-15 < 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. Taylor expanded in y around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. flip-- 81.0%

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

      3. associate-*l/ 78.1%

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

      4. metadata-eval 78.1%

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

      5. sub-neg 78.1%

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

      6. pow2 78.1%

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

      7. metadata-eval 78.1%

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

   4. Applied egg-rr 78.1%

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

      1. unpow2 78.1%

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

      2. difference-of-sqr--1 78.1%

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

      3. difference-of-sqr-1 78.1%

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

      4. metadata-eval 78.1%

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

      5. associate-*l/ 81.0%

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

      6. flip-- 98.3%

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

      7. *-commutative 98.3%

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

      8. associate-*r* 98.3%

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

      9. sub-neg 98.3%

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

      10. metadata-eval 98.3%

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

   6. Applied egg-rr 98.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-55}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{\frac{1}{x}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 6: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-54}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{\frac{3}{x}}{9}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.4e-82)
   (/ (sqrt x) (* x 3.0))
   (if (<= x 4.5e-54)
     (* 3.0 (* (sqrt x) y))
     (if (<= x 8.5e-15)
       (* (sqrt x) (+ -3.0 (/ (/ 3.0 x) 9.0)))
       (* 3.0 (* (sqrt x) (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.4e-82) {
		tmp = sqrt(x) / (x * 3.0);
	} else if (x <= 4.5e-54) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (x <= 8.5e-15) {
		tmp = sqrt(x) * (-3.0 + ((3.0 / x) / 9.0));
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.4d-82) then
        tmp = sqrt(x) / (x * 3.0d0)
    else if (x <= 4.5d-54) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (x <= 8.5d-15) then
        tmp = sqrt(x) * ((-3.0d0) + ((3.0d0 / x) / 9.0d0))
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.4e-82) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else if (x <= 4.5e-54) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (x <= 8.5e-15) {
		tmp = Math.sqrt(x) * (-3.0 + ((3.0 / x) / 9.0));
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.4e-82:
		tmp = math.sqrt(x) / (x * 3.0)
	elif x <= 4.5e-54:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif x <= 8.5e-15:
		tmp = math.sqrt(x) * (-3.0 + ((3.0 / x) / 9.0))
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.4e-82)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	elseif (x <= 4.5e-54)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (x <= 8.5e-15)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(Float64(3.0 / x) / 9.0)));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.4e-82)
		tmp = sqrt(x) / (x * 3.0);
	elseif (x <= 4.5e-54)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (x <= 8.5e-15)
		tmp = sqrt(x) * (-3.0 + ((3.0 / x) / 9.0));
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.4e-82], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-54], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-15], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(N[(3.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.40000000000000012e-82

   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. metadata-eval 99.2%

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

      2. sqrt-prod 99.4%

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

      3. *-commutative 99.4%

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

      4. pow1/2 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. unpow1/2 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. metadata-eval 74.1%

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

      3. sub-neg 74.1%

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

      4. metadata-eval 74.1%

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

      5. associate-*l* 74.1%

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

      6. *-commutative 74.1%

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

      7. associate-*l* 74.1%

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

      8. +-commutative 74.1%

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

      9. distribute-rgt-in 74.1%

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

      10. metadata-eval 74.1%

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

   8. Simplified 74.1%

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

      1. flip-+ 19.1%

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

      2. associate-*r/ 19.1%

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

      3. associate-/l* 19.1%

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

      4. *-un-lft-identity 19.1%

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

      5. associate-/l* 19.1%

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

      6. flip-+ 74.1%

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

      7. associate-*l/ 74.2%

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

      8. metadata-eval 74.2%

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

   10. Applied egg-rr 74.2%

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

   11. Taylor expanded in x around 0 74.4%

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

   if 1.40000000000000012e-82 < x < 4.4999999999999998e-54

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   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 inf 89.7%

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

   if 4.4999999999999998e-54 < x < 8.50000000000000007e-15

   1. Initial program 99.0%

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

      1. metadata-eval 99.0%

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

      2. sqrt-prod 99.5%

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

      3. *-commutative 99.5%

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

      4. pow1/2 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around 0 62.4%

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

      1. associate-*r/ 62.6%

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

      2. metadata-eval 62.6%

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

      3. sub-neg 62.6%

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

      4. metadata-eval 62.6%

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

      5. associate-*l* 62.5%

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

      6. *-commutative 62.5%

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

      7. associate-*l* 62.6%

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

      8. +-commutative 62.6%

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

      9. distribute-rgt-in 62.6%

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

      10. metadata-eval 62.6%

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

   8. Simplified 62.6%

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

      1. clear-num 62.2%

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

      2. associate-*l/ 62.3%

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

      3. metadata-eval 62.3%

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

      4. div-inv 62.7%

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

      5. metadata-eval 62.7%

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

      6. metadata-eval 62.7%

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

      7. associate-/r* 62.9%

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

      8. metadata-eval 62.9%

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

   10. Applied egg-rr 62.9%

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

   if 8.50000000000000007e-15 < 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. Taylor expanded in y around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. flip-- 81.0%

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

      3. associate-*l/ 78.1%

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

      4. metadata-eval 78.1%

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

      5. sub-neg 78.1%

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

      6. pow2 78.1%

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

      7. metadata-eval 78.1%

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

   4. Applied egg-rr 78.1%

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

      1. unpow2 78.1%

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

      2. difference-of-sqr--1 78.1%

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

      3. difference-of-sqr-1 78.1%

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

      4. metadata-eval 78.1%

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

      5. associate-*l/ 81.0%

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

      6. flip-- 98.3%

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

      7. *-commutative 98.3%

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

      8. associate-*r* 98.3%

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

      9. sub-neg 98.3%

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

      10. metadata-eval 98.3%

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

   6. Applied egg-rr 98.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-54}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{\frac{3}{x}}{9}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 7: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-54}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.65e-82)
   (/ (sqrt x) (* x 3.0))
   (if (<= x 5.5e-54)
     (* 3.0 (* (sqrt x) y))
     (if (<= x 8.6e-15)
       (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
       (* (sqrt x) (- (* 3.0 y) 3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.65e-82) {
		tmp = sqrt(x) / (x * 3.0);
	} else if (x <= 5.5e-54) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (x <= 8.6e-15) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.65d-82) then
        tmp = sqrt(x) / (x * 3.0d0)
    else if (x <= 5.5d-54) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (x <= 8.6d-15) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.65e-82) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else if (x <= 5.5e-54) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (x <= 8.6e-15) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.65e-82:
		tmp = math.sqrt(x) / (x * 3.0)
	elif x <= 5.5e-54:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif x <= 8.6e-15:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.65e-82)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	elseif (x <= 5.5e-54)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (x <= 8.6e-15)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) - 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.65e-82)
		tmp = sqrt(x) / (x * 3.0);
	elseif (x <= 5.5e-54)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (x <= 8.6e-15)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.65e-82], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-54], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.6e-15], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.65000000000000011e-82

   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. metadata-eval 99.2%

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

      2. sqrt-prod 99.4%

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

      3. *-commutative 99.4%

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

      4. pow1/2 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. unpow1/2 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. metadata-eval 74.1%

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

      3. sub-neg 74.1%

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

      4. metadata-eval 74.1%

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

      5. associate-*l* 74.1%

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

      6. *-commutative 74.1%

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

      7. associate-*l* 74.1%

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

      8. +-commutative 74.1%

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

      9. distribute-rgt-in 74.1%

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

      10. metadata-eval 74.1%

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

   8. Simplified 74.1%

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

      1. flip-+ 19.1%

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

      2. associate-*r/ 19.1%

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

      3. associate-/l* 19.1%

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

      4. *-un-lft-identity 19.1%

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

      5. associate-/l* 19.1%

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

      6. flip-+ 74.1%

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

      7. associate-*l/ 74.2%

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

      8. metadata-eval 74.2%

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

   10. Applied egg-rr 74.2%

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

   11. Taylor expanded in x around 0 74.4%

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

   if 1.65000000000000011e-82 < x < 5.50000000000000046e-54

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   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 inf 89.7%

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

   if 5.50000000000000046e-54 < x < 8.5999999999999993e-15

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.2%

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

      14. metadata-eval 99.2%

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

      15. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around 0 62.4%

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

      1. associate-*r/ 62.6%

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

      2. metadata-eval 62.6%

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

      3. sub-neg 62.6%

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

      4. metadata-eval 62.6%

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

   6. Simplified 62.6%

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

   if 8.5999999999999993e-15 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-54}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]

Alternative 8: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-55}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8.2e-83)
   (/ (sqrt x) (* x 3.0))
   (if (<= x 6.2e-55)
     (* 3.0 (* (sqrt x) y))
     (if (<= x 8.6e-15)
       (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
       (* 3.0 (* (sqrt x) (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.2e-83) {
		tmp = sqrt(x) / (x * 3.0);
	} else if (x <= 6.2e-55) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (x <= 8.6e-15) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 8.2d-83) then
        tmp = sqrt(x) / (x * 3.0d0)
    else if (x <= 6.2d-55) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (x <= 8.6d-15) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8.2e-83) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else if (x <= 6.2e-55) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (x <= 8.6e-15) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8.2e-83:
		tmp = math.sqrt(x) / (x * 3.0)
	elif x <= 6.2e-55:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif x <= 8.6e-15:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.2e-83)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	elseif (x <= 6.2e-55)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (x <= 8.6e-15)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8.2e-83)
		tmp = sqrt(x) / (x * 3.0);
	elseif (x <= 6.2e-55)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (x <= 8.6e-15)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.2e-83], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e-55], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.6e-15], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.1999999999999999e-83

   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. metadata-eval 99.2%

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

      2. sqrt-prod 99.4%

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

      3. *-commutative 99.4%

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

      4. pow1/2 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. unpow1/2 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. metadata-eval 74.1%

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

      3. sub-neg 74.1%

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

      4. metadata-eval 74.1%

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

      5. associate-*l* 74.1%

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

      6. *-commutative 74.1%

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

      7. associate-*l* 74.1%

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

      8. +-commutative 74.1%

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

      9. distribute-rgt-in 74.1%

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

      10. metadata-eval 74.1%

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

   8. Simplified 74.1%

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

      1. flip-+ 19.1%

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

      2. associate-*r/ 19.1%

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

      3. associate-/l* 19.1%

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

      4. *-un-lft-identity 19.1%

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

      5. associate-/l* 19.1%

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

      6. flip-+ 74.1%

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

      7. associate-*l/ 74.2%

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

      8. metadata-eval 74.2%

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

   10. Applied egg-rr 74.2%

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

   11. Taylor expanded in x around 0 74.4%

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

   if 8.1999999999999999e-83 < x < 6.19999999999999993e-55

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   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 inf 89.7%

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

   if 6.19999999999999993e-55 < x < 8.5999999999999993e-15

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.2%

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

      14. metadata-eval 99.2%

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

      15. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around 0 62.4%

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

      1. associate-*r/ 62.6%

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

      2. metadata-eval 62.6%

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

      3. sub-neg 62.6%

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

      4. metadata-eval 62.6%

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

   6. Simplified 62.6%

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

   if 8.5999999999999993e-15 < 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. Taylor expanded in y around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. flip-- 81.0%

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

      3. associate-*l/ 78.1%

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

      4. metadata-eval 78.1%

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

      5. sub-neg 78.1%

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

      6. pow2 78.1%

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

      7. metadata-eval 78.1%

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

   4. Applied egg-rr 78.1%

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

      1. unpow2 78.1%

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

      2. difference-of-sqr--1 78.1%

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

      3. difference-of-sqr-1 78.1%

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

      4. metadata-eval 78.1%

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

      5. associate-*l/ 81.0%

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

      6. flip-- 98.3%

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

      7. *-commutative 98.3%

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

      8. associate-*r* 98.3%

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

      9. sub-neg 98.3%

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

      10. metadata-eval 98.3%

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

   6. Applied egg-rr 98.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-55}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 9: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+146} \lor \neg \left(y \leq 1.5 \cdot 10^{+106}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.5e+146) (not (<= y 1.5e+106)))
   (* 3.0 (* (sqrt x) y))
   (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.5e+146) || !(y <= 1.5e+106)) {
		tmp = 3.0 * (sqrt(x) * y);
	} else {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.5d+146)) .or. (.not. (y <= 1.5d+106))) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.5e+146) || !(y <= 1.5e+106)) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.5e+146) or not (y <= 1.5e+106):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.5e+146) || !(y <= 1.5e+106))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.5e+146) || ~((y <= 1.5e+106)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.5e+146], N[Not[LessEqual[y, 1.5e+106]], $MachinePrecision]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.49999999999999983e146 or 1.5e106 < 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%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.4%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 85.8%

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

   if -7.49999999999999983e146 < y < 1.5e106

   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%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 84.1%

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

      1. associate-*r/ 84.1%

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

      2. metadata-eval 84.1%

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

      3. sub-neg 84.1%

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

      4. metadata-eval 84.1%

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

   6. Simplified 84.1%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+146} \lor \neg \left(y \leq 1.5 \cdot 10^{+106}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \end{array} \]

Alternative 10: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 4.1 \cdot 10^{-17}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 4.1e-17)))
   (* 3.0 (* (sqrt x) y))
   (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 4.1e-17)) {
		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 <= 4.1d-17))) 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 <= 4.1e-17)) {
		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 <= 4.1e-17):
		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 <= 4.1e-17))
		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 <= 4.1e-17)))
		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, 4.1e-17]], $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 4.1000000000000001e-17 < 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%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 69.4%

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

   if -1 < y < 4.1000000000000001e-17

   1. Initial program 99.3%

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

   2. Taylor expanded in y around inf 59.3%

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

   3. Taylor expanded in y around 0 57.6%

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

      1. *-commutative 57.6%

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

   5. Simplified 57.6%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 4.1 \cdot 10^{-17}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 11: 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 99.4%

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

2. Final simplification 99.4%

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

Alternative 12: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

   7. metadata-eval 99.3%

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

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. metadata-eval 99.4%

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

   2. sqrt-prod 98.8%

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

   3. *-commutative 98.8%

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

   4. pow1/2 98.8%

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

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 98.8%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 13: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

   7. metadata-eval 99.3%

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

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. Final simplification 99.3%

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

Alternative 14: 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 99.4%

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

2. Taylor expanded in y around inf 66.4%

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

3. Taylor expanded in y around 0 25.9%

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

   1. *-commutative 25.9%

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

5. Simplified 25.9%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 3.2%

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

   3. swap-sqr 3.2%

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

   4. add-sqr-sqrt 3.2%

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

   5. metadata-eval 3.2%

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

7. Applied egg-rr 3.2%

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

8. Final simplification 3.2%

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

Alternative 15: 25.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot -3 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in y around inf 66.4%

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

3. Taylor expanded in y around 0 25.9%

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

   1. *-commutative 25.9%

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

5. Simplified 25.9%

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

6. Final simplification 25.9%

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

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 2023336 
(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)))