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

Percentage Accurate: 99.4% → 99.4%

Time: 10.2s

Alternatives: 12

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

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

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) * Float64(fma(3.0, y, -3.0) - Float64(-0.3333333333333333 / x)))
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{if}\;y \leq -14.6:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (/ 0.3333333333333333 x))))
   (if (<= y -14.6)
     (* 3.0 (* (sqrt x) y))
     (if (<= y -6.5e-260)
       t_0
       (if (<= y 2.6e-29)
         (* (sqrt x) -3.0)
         (if (<= y 1.65e+38) t_0 (* (sqrt x) (* 3.0 y))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (y <= -14.6) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= -6.5e-260) {
		tmp = t_0;
	} else if (y <= 2.6e-29) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 1.65e+38) {
		tmp = t_0;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * (0.3333333333333333d0 / x)
    if (y <= (-14.6d0)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= (-6.5d-260)) then
        tmp = t_0
    else if (y <= 2.6d-29) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 1.65d+38) then
        tmp = t_0
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (y <= -14.6) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= -6.5e-260) {
		tmp = t_0;
	} else if (y <= 2.6e-29) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 1.65e+38) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * (0.3333333333333333 / x)
	tmp = 0
	if y <= -14.6:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= -6.5e-260:
		tmp = t_0
	elif y <= 2.6e-29:
		tmp = math.sqrt(x) * -3.0
	elif y <= 1.65e+38:
		tmp = t_0
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * Float64(0.3333333333333333 / x))
	tmp = 0.0
	if (y <= -14.6)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= -6.5e-260)
		tmp = t_0;
	elseif (y <= 2.6e-29)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 1.65e+38)
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * (0.3333333333333333 / x);
	tmp = 0.0;
	if (y <= -14.6)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= -6.5e-260)
		tmp = t_0;
	elseif (y <= 2.6e-29)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 1.65e+38)
		tmp = t_0;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -14.6], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -6.5e-260], t$95$0, If[LessEqual[y, 2.6e-29], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 1.65e+38], t$95$0, N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -14.5999999999999996

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 77.8%

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

   if -14.5999999999999996 < y < -6.50000000000000002e-260 or 2.6000000000000002e-29 < y < 1.65e38

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 57.2%

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

   if -6.50000000000000002e-260 < y < 2.6000000000000002e-29

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. metadata-eval 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. add-cube-cbrt 98.1%

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

      2. pow3 98.0%

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

      3. +-commutative 98.0%

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

   8. Applied egg-rr 98.0%

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

   9. Taylor expanded in x around inf 60.4%

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

       1. *-commutative 60.4%

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

   11. Simplified 60.4%

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

   if 1.65e38 < y

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 79.3%

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

      1. associate-*r* 79.4%

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

      2. *-commutative 79.4%

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

   6. Simplified 79.4%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14.6:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-260}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]

Alternative 3: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{if}\;y \leq -14.6:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (/ 0.3333333333333333 x))))
   (if (<= y -14.6)
     (* 3.0 (* (sqrt x) y))
     (if (<= y -2.6e-259)
       t_0
       (if (<= y 3.2e-29)
         (* (sqrt x) -3.0)
         (if (<= y 1.65e+38) t_0 (* y (* (sqrt x) 3.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (y <= -14.6) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= -2.6e-259) {
		tmp = t_0;
	} else if (y <= 3.2e-29) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 1.65e+38) {
		tmp = t_0;
	} else {
		tmp = y * (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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * (0.3333333333333333d0 / x)
    if (y <= (-14.6d0)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= (-2.6d-259)) then
        tmp = t_0
    else if (y <= 3.2d-29) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 1.65d+38) then
        tmp = t_0
    else
        tmp = y * (sqrt(x) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (y <= -14.6) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= -2.6e-259) {
		tmp = t_0;
	} else if (y <= 3.2e-29) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 1.65e+38) {
		tmp = t_0;
	} else {
		tmp = y * (Math.sqrt(x) * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * (0.3333333333333333 / x)
	tmp = 0
	if y <= -14.6:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= -2.6e-259:
		tmp = t_0
	elif y <= 3.2e-29:
		tmp = math.sqrt(x) * -3.0
	elif y <= 1.65e+38:
		tmp = t_0
	else:
		tmp = y * (math.sqrt(x) * 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * Float64(0.3333333333333333 / x))
	tmp = 0.0
	if (y <= -14.6)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= -2.6e-259)
		tmp = t_0;
	elseif (y <= 3.2e-29)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 1.65e+38)
		tmp = t_0;
	else
		tmp = Float64(y * Float64(sqrt(x) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * (0.3333333333333333 / x);
	tmp = 0.0;
	if (y <= -14.6)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= -2.6e-259)
		tmp = t_0;
	elseif (y <= 3.2e-29)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 1.65e+38)
		tmp = t_0;
	else
		tmp = y * (sqrt(x) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -14.6], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.6e-259], t$95$0, If[LessEqual[y, 3.2e-29], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 1.65e+38], t$95$0, N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -14.5999999999999996

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 77.8%

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

   if -14.5999999999999996 < y < -2.60000000000000001e-259 or 3.2e-29 < y < 1.65e38

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 57.2%

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

   if -2.60000000000000001e-259 < y < 3.2e-29

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-*r/ 99.6%

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

      4. metadata-eval 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. add-cube-cbrt 98.1%

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

      2. pow3 98.0%

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

      3. +-commutative 98.0%

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

   8. Applied egg-rr 98.0%

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

   9. Taylor expanded in x around inf 60.4%

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

       1. *-commutative 60.4%

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

   11. Simplified 60.4%

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

   if 1.65e38 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.6%

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

      11. distribute-neg-frac 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r/ 99.7%

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

      14. associate-/l/ 99.7%

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

      15. associate-/r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 79.5%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14.6:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]

Alternative 4: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11 \lor \neg \left(y \leq 1.65 \cdot 10^{+38}\right):\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{1}{x \cdot 3} - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -11.0) (not (<= y 1.65e+38)))
   (* (* (sqrt x) 3.0) (- y 1.0))
   (* (sqrt x) (- (/ 1.0 (* x 3.0)) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -11.0) || !(y <= 1.65e+38)) {
		tmp = (sqrt(x) * 3.0) * (y - 1.0);
	} else {
		tmp = sqrt(x) * ((1.0 / (x * 3.0)) - 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 <= (-11.0d0)) .or. (.not. (y <= 1.65d+38))) then
        tmp = (sqrt(x) * 3.0d0) * (y - 1.0d0)
    else
        tmp = sqrt(x) * ((1.0d0 / (x * 3.0d0)) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -11.0) || !(y <= 1.65e+38)) {
		tmp = (Math.sqrt(x) * 3.0) * (y - 1.0);
	} else {
		tmp = Math.sqrt(x) * ((1.0 / (x * 3.0)) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -11.0) or not (y <= 1.65e+38):
		tmp = (math.sqrt(x) * 3.0) * (y - 1.0)
	else:
		tmp = math.sqrt(x) * ((1.0 / (x * 3.0)) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -11.0) || !(y <= 1.65e+38))
		tmp = Float64(Float64(sqrt(x) * 3.0) * Float64(y - 1.0));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(1.0 / Float64(x * 3.0)) - 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -11.0) || ~((y <= 1.65e+38)))
		tmp = (sqrt(x) * 3.0) * (y - 1.0);
	else
		tmp = sqrt(x) * ((1.0 / (x * 3.0)) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -11.0], N[Not[LessEqual[y, 1.65e+38]], $MachinePrecision]], N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(1.0 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -11 or 1.65e38 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.6%

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

      11. distribute-neg-frac 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r/ 99.6%

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

      14. associate-/l/ 99.6%

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

      15. associate-/r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 80.0%

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

   if -11 < y < 1.65e38

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 96.9%

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

      1. div-inv 96.9%

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

      2. clear-num 96.8%

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

      3. div-inv 96.9%

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

      4. metadata-eval 96.9%

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

   6. Applied egg-rr 96.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11 \lor \neg \left(y \leq 1.65 \cdot 10^{+38}\right):\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{1}{x \cdot 3} - 3\right)\\ \end{array} \]

Alternative 5: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \lor \neg \left(y \leq 1.65 \cdot 10^{+38}\right):\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.2) (not (<= y 1.65e+38)))
   (* (* (sqrt x) 3.0) (- y 1.0))
   (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.2) || !(y <= 1.65e+38)) {
		tmp = (sqrt(x) * 3.0) * (y - 1.0);
	} 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 <= (-6.2d0)) .or. (.not. (y <= 1.65d+38))) then
        tmp = (sqrt(x) * 3.0d0) * (y - 1.0d0)
    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 <= -6.2) || !(y <= 1.65e+38)) {
		tmp = (Math.sqrt(x) * 3.0) * (y - 1.0);
	} else {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.2) or not (y <= 1.65e+38):
		tmp = (math.sqrt(x) * 3.0) * (y - 1.0)
	else:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.2) || !(y <= 1.65e+38))
		tmp = Float64(Float64(sqrt(x) * 3.0) * Float64(y - 1.0));
	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 <= -6.2) || ~((y <= 1.65e+38)))
		tmp = (sqrt(x) * 3.0) * (y - 1.0);
	else
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.2], N[Not[LessEqual[y, 1.65e+38]], $MachinePrecision]], N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y - 1.0), $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 < -6.20000000000000018 or 1.65e38 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.6%

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

      11. distribute-neg-frac 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r/ 99.6%

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

      14. associate-/l/ 99.6%

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

      15. associate-/r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 80.0%

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

   if -6.20000000000000018 < y < 1.65e38

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 96.9%

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

      1. *-commutative 96.9%

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

      2. sub-neg 96.9%

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

      3. associate-*r/ 96.9%

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

      4. metadata-eval 96.9%

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

      5. metadata-eval 96.9%

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

   6. Simplified 96.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \lor \neg \left(y \leq 1.65 \cdot 10^{+38}\right):\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \end{array} \]

Alternative 6: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -14.6:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -14.6)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 1.65e+38)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (* y (* (sqrt x) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -14.6) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 1.65e+38) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * (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 <= (-14.6d0)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 1.65d+38) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = y * (sqrt(x) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -14.6) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 1.65e+38) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * (Math.sqrt(x) * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -14.6:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 1.65e+38:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = y * (math.sqrt(x) * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -14.6)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 1.65e+38)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(y * Float64(sqrt(x) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -14.6)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 1.65e+38)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = y * (sqrt(x) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -14.6], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e+38], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -14.5999999999999996

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 77.8%

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

   if -14.5999999999999996 < y < 1.65e38

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 96.9%

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

      1. *-commutative 96.9%

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

      2. sub-neg 96.9%

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

      3. associate-*r/ 96.9%

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

      4. metadata-eval 96.9%

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

      5. metadata-eval 96.9%

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

   6. Simplified 96.9%

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

   if 1.65e38 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.6%

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

      11. distribute-neg-frac 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r/ 99.7%

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

      14. associate-/l/ 99.7%

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

      15. associate-/r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 79.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14.6:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]

Alternative 7: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.5)
   (* (sqrt x) (- (* 3.0 y) 3.0))
   (if (<= y 1.65e+38)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (* y (* (sqrt x) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.5) {
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	} else if (y <= 1.65e+38) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * (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 <= (-3.5d0)) then
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    else if (y <= 1.65d+38) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = y * (sqrt(x) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.5) {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	} else if (y <= 1.65e+38) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * (Math.sqrt(x) * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.5:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	elif y <= 1.65e+38:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = y * (math.sqrt(x) * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.5)
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) - 3.0));
	elseif (y <= 1.65e+38)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(y * Float64(sqrt(x) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.5)
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	elseif (y <= 1.65e+38)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = y * (sqrt(x) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.5], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e+38], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 80.4%

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

   if -3.5 < y < 1.65e38

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 96.9%

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

      1. *-commutative 96.9%

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

      2. sub-neg 96.9%

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

      3. associate-*r/ 96.9%

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

      4. metadata-eval 96.9%

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

      5. metadata-eval 96.9%

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

   6. Simplified 96.9%

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

   if 1.65e38 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.6%

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

      11. distribute-neg-frac 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r/ 99.7%

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

      14. associate-/l/ 99.7%

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

      15. associate-/r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 79.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]

Alternative 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. associate--l+ 99.5%

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

   3. distribute-lft-in 99.5%

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

   4. +-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*r* 99.5%

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

   7. cancel-sign-sub 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-*r* 99.5%

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

   10. *-commutative 99.5%

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

   11. distribute-rgt-out-- 99.5%

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

   12. distribute-lft-neg-in 99.5%

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

   13. cancel-sign-sub 99.5%

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

   14. +-commutative 99.5%

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

   15. *-commutative 99.5%

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

   16. distribute-rgt-in 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 9: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-22} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.4e-22) (not (<= y 1.0)))
   (* 3.0 (* (sqrt x) y))
   (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.4e-22) || !(y <= 1.0)) {
		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 <= (-7.4d-22)) .or. (.not. (y <= 1.0d0))) 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 <= -7.4e-22) || !(y <= 1.0)) {
		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 <= -7.4e-22) or not (y <= 1.0):
		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 <= -7.4e-22) || !(y <= 1.0))
		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 <= -7.4e-22) || ~((y <= 1.0)))
		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, -7.4e-22], N[Not[LessEqual[y, 1.0]], $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 < -7.4e-22 or 1 < y

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 73.6%

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

   if -7.4e-22 < y < 1

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 98.4%

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

      1. *-commutative 98.4%

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

      2. sub-neg 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

      5. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

      1. add-cube-cbrt 97.0%

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

      2. pow3 97.0%

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

      3. +-commutative 97.0%

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

   8. Applied egg-rr 97.0%

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

   9. Taylor expanded in x around inf 53.4%

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

       1. *-commutative 53.4%

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

   11. Simplified 53.4%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-22} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 10: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-22}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.4e-22)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 1.0) (* (sqrt x) -3.0) (* (sqrt x) (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.4e-22) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 1.0) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.4d-22)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 1.0d0) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.4e-22) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 1.0) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.4e-22:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 1.0:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.4e-22)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 1.0)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.4e-22)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 1.0)
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.4e-22], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.4e-22

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 74.7%

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

   if -7.4e-22 < y < 1

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 98.4%

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

      1. *-commutative 98.4%

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

      2. sub-neg 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

      5. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

      1. add-cube-cbrt 97.0%

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

      2. pow3 97.0%

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

      3. +-commutative 97.0%

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

   8. Applied egg-rr 97.0%

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

   9. Taylor expanded in x around inf 53.4%

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

       1. *-commutative 53.4%

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

   11. Simplified 53.4%

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

   if 1 < y

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 72.2%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

   6. Simplified 72.3%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-22}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]

Alternative 11: 3.2% 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 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 59.0%

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

   1. *-commutative 59.0%

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

   2. sub-neg 59.0%

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

   3. associate-*r/ 59.0%

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

   4. metadata-eval 59.0%

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

   5. metadata-eval 59.0%

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

6. Simplified 59.0%

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

   1. add-cube-cbrt 58.2%

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

   2. pow3 58.3%

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

   3. +-commutative 58.3%

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

8. Applied egg-rr 58.3%

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

9. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{-3 \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{x}\right)\right)} \]
10. Step-by-step derivation

    1. associate-*r* 0.0%

       \[\leadsto \color{blue}{\left(-3 \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{x}\right)} \]

    2. unpow2 0.0%

       \[\leadsto \left(-3 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{x}\right) \]

    3. rem-square-sqrt 3.1%

       \[\leadsto \left(-3 \cdot \color{blue}{-1}\right) \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{x}\right) \]

    4. metadata-eval 3.1%

       \[\leadsto \color{blue}{3} \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{x}\right) \]

    5. pow-base-1 3.1%

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

    6. *-lft-identity 3.1%

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

11. Simplified 3.1%

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

12. Final simplification 3.1%

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

Alternative 12: 26.3% 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 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 59.0%

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

   1. *-commutative 59.0%

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

   2. sub-neg 59.0%

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

   3. associate-*r/ 59.0%

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

   4. metadata-eval 59.0%

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

   5. metadata-eval 59.0%

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

6. Simplified 59.0%

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

   1. add-cube-cbrt 58.2%

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

   2. pow3 58.3%

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

   3. +-commutative 58.3%

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

8. Applied egg-rr 58.3%

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

9. Taylor expanded in x around inf 25.8%

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

    1. *-commutative 25.8%

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

11. Simplified 25.8%

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

12. Final simplification 25.8%

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

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 2023229 
(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)))