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

Percentage Accurate: 99.4% → 99.4%

Time: 8.4s

Alternatives: 13

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Final simplification 99.4%

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \]
4. Add Preprocessing

Alternative 2: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\left(\left(y - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_0 -20.0)
     (* (* (- y 1.0) (sqrt x)) 3.0)
     (if (<= t_0 2e+153)
       (* (sqrt (pow x -1.0)) 0.3333333333333333)
       (* (* 3.0 y) (sqrt x))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = ((y - 1.0) * sqrt(x)) * 3.0;
	} else if (t_0 <= 2e+153) {
		tmp = sqrt(pow(x, -1.0)) * 0.3333333333333333;
	} else {
		tmp = (3.0 * y) * sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (3.0d0 * sqrt(x)) * ((y + ((x * 9.0d0) ** (-1.0d0))) - 1.0d0)
    if (t_0 <= (-20.0d0)) then
        tmp = ((y - 1.0d0) * sqrt(x)) * 3.0d0
    else if (t_0 <= 2d+153) then
        tmp = sqrt((x ** (-1.0d0))) * 0.3333333333333333d0
    else
        tmp = (3.0d0 * y) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (3.0 * Math.sqrt(x)) * ((y + Math.pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = ((y - 1.0) * Math.sqrt(x)) * 3.0;
	} else if (t_0 <= 2e+153) {
		tmp = Math.sqrt(Math.pow(x, -1.0)) * 0.3333333333333333;
	} else {
		tmp = (3.0 * y) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (3.0 * math.sqrt(x)) * ((y + math.pow((x * 9.0), -1.0)) - 1.0)
	tmp = 0
	if t_0 <= -20.0:
		tmp = ((y - 1.0) * math.sqrt(x)) * 3.0
	elif t_0 <= 2e+153:
		tmp = math.sqrt(math.pow(x, -1.0)) * 0.3333333333333333
	else:
		tmp = (3.0 * y) * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = Float64(Float64(Float64(y - 1.0) * sqrt(x)) * 3.0);
	elseif (t_0 <= 2e+153)
		tmp = Float64(sqrt((x ^ -1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(3.0 * y) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (3.0 * sqrt(x)) * ((y + ((x * 9.0) ^ -1.0)) - 1.0);
	tmp = 0.0;
	if (t_0 <= -20.0)
		tmp = ((y - 1.0) * sqrt(x)) * 3.0;
	elseif (t_0 <= 2e+153)
		tmp = sqrt((x ^ -1.0)) * 0.3333333333333333;
	else
		tmp = (3.0 * y) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], N[(N[(N[(y - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+153], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -20

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 97.3

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

   7. Applied rewrites 97.3%

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

   if -20 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 2e153

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt{\frac{1}{x}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if 2e153 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -20:\\ \;\;\;\;\left(\left(y - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(\frac{0.1111111111111111}{x} - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_0 -2e+40)
     (* (* (- y 1.0) (sqrt x)) 3.0)
     (if (<= t_0 2e+153)
       (* (* (- (/ 0.1111111111111111 x) 1.0) 3.0) (sqrt x))
       (* (* 3.0 y) (sqrt x))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -2e+40) {
		tmp = ((y - 1.0) * sqrt(x)) * 3.0;
	} else if (t_0 <= 2e+153) {
		tmp = (((0.1111111111111111 / x) - 1.0) * 3.0) * sqrt(x);
	} else {
		tmp = (3.0 * y) * sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (3.0d0 * sqrt(x)) * ((y + ((x * 9.0d0) ** (-1.0d0))) - 1.0d0)
    if (t_0 <= (-2d+40)) then
        tmp = ((y - 1.0d0) * sqrt(x)) * 3.0d0
    else if (t_0 <= 2d+153) then
        tmp = (((0.1111111111111111d0 / x) - 1.0d0) * 3.0d0) * sqrt(x)
    else
        tmp = (3.0d0 * y) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (3.0 * Math.sqrt(x)) * ((y + Math.pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -2e+40) {
		tmp = ((y - 1.0) * Math.sqrt(x)) * 3.0;
	} else if (t_0 <= 2e+153) {
		tmp = (((0.1111111111111111 / x) - 1.0) * 3.0) * Math.sqrt(x);
	} else {
		tmp = (3.0 * y) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (3.0 * math.sqrt(x)) * ((y + math.pow((x * 9.0), -1.0)) - 1.0)
	tmp = 0
	if t_0 <= -2e+40:
		tmp = ((y - 1.0) * math.sqrt(x)) * 3.0
	elif t_0 <= 2e+153:
		tmp = (((0.1111111111111111 / x) - 1.0) * 3.0) * math.sqrt(x)
	else:
		tmp = (3.0 * y) * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_0 <= -2e+40)
		tmp = Float64(Float64(Float64(y - 1.0) * sqrt(x)) * 3.0);
	elseif (t_0 <= 2e+153)
		tmp = Float64(Float64(Float64(Float64(0.1111111111111111 / x) - 1.0) * 3.0) * sqrt(x));
	else
		tmp = Float64(Float64(3.0 * y) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (3.0 * sqrt(x)) * ((y + ((x * 9.0) ^ -1.0)) - 1.0);
	tmp = 0.0;
	if (t_0 <= -2e+40)
		tmp = ((y - 1.0) * sqrt(x)) * 3.0;
	elseif (t_0 <= 2e+153)
		tmp = (((0.1111111111111111 / x) - 1.0) * 3.0) * sqrt(x);
	else
		tmp = (3.0 * y) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+40], N[(N[(N[(y - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+153], N[(N[(N[(N[(0.1111111111111111 / x), $MachinePrecision] - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -2.00000000000000006e40

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if -2.00000000000000006e40 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 2e153

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{9} \cdot \frac{1}{x} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if 2e153 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(\frac{0.1111111111111111}{x} - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\left(\left(y - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_0 -20.0)
     (* (* (- y 1.0) (sqrt x)) 3.0)
     (if (<= t_0 2e+153)
       (/ (* 0.3333333333333333 (sqrt x)) x)
       (* (* 3.0 y) (sqrt x))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = ((y - 1.0) * sqrt(x)) * 3.0;
	} else if (t_0 <= 2e+153) {
		tmp = (0.3333333333333333 * sqrt(x)) / x;
	} else {
		tmp = (3.0 * y) * sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (3.0d0 * sqrt(x)) * ((y + ((x * 9.0d0) ** (-1.0d0))) - 1.0d0)
    if (t_0 <= (-20.0d0)) then
        tmp = ((y - 1.0d0) * sqrt(x)) * 3.0d0
    else if (t_0 <= 2d+153) then
        tmp = (0.3333333333333333d0 * sqrt(x)) / x
    else
        tmp = (3.0d0 * y) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (3.0 * Math.sqrt(x)) * ((y + Math.pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = ((y - 1.0) * Math.sqrt(x)) * 3.0;
	} else if (t_0 <= 2e+153) {
		tmp = (0.3333333333333333 * Math.sqrt(x)) / x;
	} else {
		tmp = (3.0 * y) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (3.0 * math.sqrt(x)) * ((y + math.pow((x * 9.0), -1.0)) - 1.0)
	tmp = 0
	if t_0 <= -20.0:
		tmp = ((y - 1.0) * math.sqrt(x)) * 3.0
	elif t_0 <= 2e+153:
		tmp = (0.3333333333333333 * math.sqrt(x)) / x
	else:
		tmp = (3.0 * y) * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = Float64(Float64(Float64(y - 1.0) * sqrt(x)) * 3.0);
	elseif (t_0 <= 2e+153)
		tmp = Float64(Float64(0.3333333333333333 * sqrt(x)) / x);
	else
		tmp = Float64(Float64(3.0 * y) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (3.0 * sqrt(x)) * ((y + ((x * 9.0) ^ -1.0)) - 1.0);
	tmp = 0.0;
	if (t_0 <= -20.0)
		tmp = ((y - 1.0) * sqrt(x)) * 3.0;
	elseif (t_0 <= 2e+153)
		tmp = (0.3333333333333333 * sqrt(x)) / x;
	else
		tmp = (3.0 * y) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], N[(N[(N[(y - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+153], N[(N[(0.3333333333333333 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -20

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 97.3

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

   7. Applied rewrites 97.3%

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

   if -20 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 2e153

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.3

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 95.0

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

   7. Applied rewrites 95.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt{x}}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.2%

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

   if 2e153 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -20:\\ \;\;\;\;\left(\left(y - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(y - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

Alternative 6: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + N[(N[(0.3333333333333333 / x), $MachinePrecision] - 3.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

      \[\leadsto \left(\color{blue}{{x}^{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. sqr-pow N/A

      \[\leadsto \left(\color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   9. metadata-eval N/A

      \[\leadsto \left({x}^{\color{blue}{\frac{1}{4}}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   11. lower-pow.f64 N/A

       \[\leadsto \left({x}^{\frac{1}{4}} \cdot \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   12. metadata-eval 99.1

       \[\leadsto \left({x}^{0.25} \cdot \left({x}^{\color{blue}{0.25}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left({x}^{0.25} \cdot \left({x}^{0.25} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\frac{1}{4}} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left({x}^{\frac{1}{4}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto \left(\left(\color{blue}{{x}^{\frac{1}{4}}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   6. lift-pow.f64 N/A

      \[\leadsto \left(\left({x}^{\frac{1}{4}} \cdot \color{blue}{{x}^{\frac{1}{4}}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   7. pow-prod-up N/A

      \[\leadsto \left(\color{blue}{{x}^{\left(\frac{1}{4} + \frac{1}{4}\right)}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   8. metadata-eval N/A

      \[\leadsto \left({x}^{\color{blue}{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   9. pow1/2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate--l+ N/A

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

   15. distribute-rgt-in N/A

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

6. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around inf

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

    1. lower--.f64 N/A

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

    4. lower-/.f64 99.4

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

11. Applied rewrites 99.4%

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

Alternative 7: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-5} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.8e-5) (not (<= y 1.0)))
   (* (* 3.0 y) (sqrt x))
   (* -3.0 (sqrt x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.8e-5) || !(y <= 1.0)) {
		tmp = (3.0 * y) * sqrt(x);
	} else {
		tmp = -3.0 * sqrt(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 <= (-8.8d-5)) .or. (.not. (y <= 1.0d0))) then
        tmp = (3.0d0 * y) * sqrt(x)
    else
        tmp = (-3.0d0) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8.8e-5) || !(y <= 1.0)) {
		tmp = (3.0 * y) * Math.sqrt(x);
	} else {
		tmp = -3.0 * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.8e-5) or not (y <= 1.0):
		tmp = (3.0 * y) * math.sqrt(x)
	else:
		tmp = -3.0 * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.8e-5) || !(y <= 1.0))
		tmp = Float64(Float64(3.0 * y) * sqrt(x));
	else
		tmp = Float64(-3.0 * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8.8e-5) || ~((y <= 1.0)))
		tmp = (3.0 * y) * sqrt(x);
	else
		tmp = -3.0 * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.8e-5], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.7999999999999998e-5 or 1 < 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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 72.1%

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

   if -8.7999999999999998e-5 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{{x}^{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. sqr-pow N/A

         \[\leadsto \left(\color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{\color{blue}{\frac{1}{4}}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \left({x}^{\frac{1}{4}} \cdot \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      12. metadata-eval 99.1

          \[\leadsto \left({x}^{0.25} \cdot \left({x}^{\color{blue}{0.25}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left({x}^{0.25} \cdot \left({x}^{0.25} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\frac{1}{4}} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({x}^{\frac{1}{4}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(\color{blue}{{x}^{\frac{1}{4}}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left({x}^{\frac{1}{4}} \cdot \color{blue}{{x}^{\frac{1}{4}}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      7. pow-prod-up N/A

         \[\leadsto \left(\color{blue}{{x}^{\left(\frac{1}{4} + \frac{1}{4}\right)}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({x}^{\color{blue}{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      9. pow1/2 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate--l+ N/A

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

      15. distribute-rgt-in N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 51.5

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

   9. Applied rewrites 51.5%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 51.2%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-5} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.8e-5)
   (* (* (sqrt x) y) 3.0)
   (if (<= y 1.0) (* -3.0 (sqrt x)) (* (* 3.0 y) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.8e-5) {
		tmp = (sqrt(x) * y) * 3.0;
	} else if (y <= 1.0) {
		tmp = -3.0 * sqrt(x);
	} else {
		tmp = (3.0 * y) * sqrt(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 <= (-8.8d-5)) then
        tmp = (sqrt(x) * y) * 3.0d0
    else if (y <= 1.0d0) then
        tmp = (-3.0d0) * sqrt(x)
    else
        tmp = (3.0d0 * y) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.8e-5) {
		tmp = (Math.sqrt(x) * y) * 3.0;
	} else if (y <= 1.0) {
		tmp = -3.0 * Math.sqrt(x);
	} else {
		tmp = (3.0 * y) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.8e-5:
		tmp = (math.sqrt(x) * y) * 3.0
	elif y <= 1.0:
		tmp = -3.0 * math.sqrt(x)
	else:
		tmp = (3.0 * y) * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.8e-5)
		tmp = Float64(Float64(sqrt(x) * y) * 3.0);
	elseif (y <= 1.0)
		tmp = Float64(-3.0 * sqrt(x));
	else
		tmp = Float64(Float64(3.0 * y) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.8e-5)
		tmp = (sqrt(x) * y) * 3.0;
	elseif (y <= 1.0)
		tmp = -3.0 * sqrt(x);
	else
		tmp = (3.0 * y) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.8e-5], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[y, 1.0], N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.7999999999999998e-5

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -8.7999999999999998e-5 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{{x}^{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. sqr-pow N/A

         \[\leadsto \left(\color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{\color{blue}{\frac{1}{4}}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \left({x}^{\frac{1}{4}} \cdot \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      12. metadata-eval 99.1

          \[\leadsto \left({x}^{0.25} \cdot \left({x}^{\color{blue}{0.25}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left({x}^{0.25} \cdot \left({x}^{0.25} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\frac{1}{4}} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({x}^{\frac{1}{4}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(\color{blue}{{x}^{\frac{1}{4}}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left({x}^{\frac{1}{4}} \cdot \color{blue}{{x}^{\frac{1}{4}}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      7. pow-prod-up N/A

         \[\leadsto \left(\color{blue}{{x}^{\left(\frac{1}{4} + \frac{1}{4}\right)}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({x}^{\color{blue}{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      9. pow1/2 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate--l+ N/A

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

      15. distribute-rgt-in N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 51.5

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

   9. Applied rewrites 51.5%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 51.2%

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

   if 1 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 74.1%

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

Alternative 9: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-5}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.8e-5)
   (* (* 3.0 (sqrt x)) y)
   (if (<= y 1.0) (* -3.0 (sqrt x)) (* (* 3.0 y) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.8e-5) {
		tmp = (3.0 * sqrt(x)) * y;
	} else if (y <= 1.0) {
		tmp = -3.0 * sqrt(x);
	} else {
		tmp = (3.0 * y) * sqrt(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 <= (-8.8d-5)) then
        tmp = (3.0d0 * sqrt(x)) * y
    else if (y <= 1.0d0) then
        tmp = (-3.0d0) * sqrt(x)
    else
        tmp = (3.0d0 * y) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.8e-5) {
		tmp = (3.0 * Math.sqrt(x)) * y;
	} else if (y <= 1.0) {
		tmp = -3.0 * Math.sqrt(x);
	} else {
		tmp = (3.0 * y) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.8e-5:
		tmp = (3.0 * math.sqrt(x)) * y
	elif y <= 1.0:
		tmp = -3.0 * math.sqrt(x)
	else:
		tmp = (3.0 * y) * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.8e-5)
		tmp = Float64(Float64(3.0 * sqrt(x)) * y);
	elseif (y <= 1.0)
		tmp = Float64(-3.0 * sqrt(x));
	else
		tmp = Float64(Float64(3.0 * y) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.8e-5)
		tmp = (3.0 * sqrt(x)) * y;
	elseif (y <= 1.0)
		tmp = -3.0 * sqrt(x);
	else
		tmp = (3.0 * y) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.8e-5], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.0], N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.7999999999999998e-5

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 71.0%

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

   if -8.7999999999999998e-5 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \left(\color{blue}{{x}^{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. sqr-pow N/A

         \[\leadsto \left(\color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{\color{blue}{\frac{1}{4}}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \left({x}^{\frac{1}{4}} \cdot \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      12. metadata-eval 99.1

          \[\leadsto \left({x}^{0.25} \cdot \left({x}^{\color{blue}{0.25}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left({x}^{0.25} \cdot \left({x}^{0.25} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\frac{1}{4}} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({x}^{\frac{1}{4}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(\color{blue}{{x}^{\frac{1}{4}}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left({x}^{\frac{1}{4}} \cdot \color{blue}{{x}^{\frac{1}{4}}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      7. pow-prod-up N/A

         \[\leadsto \left(\color{blue}{{x}^{\left(\frac{1}{4} + \frac{1}{4}\right)}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({x}^{\color{blue}{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      9. pow1/2 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate--l+ N/A

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

      15. distribute-rgt-in N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 51.5

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

   9. Applied rewrites 51.5%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 51.2%

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

   if 1 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 74.1%

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

Alternative 10: 62.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(y - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around inf

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

   1. lower--.f64 63.1

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

7. Applied rewrites 63.1%

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

Alternative 11: 62.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around inf

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

   1. lower--.f64 63.0

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

7. Applied rewrites 63.0%

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

Alternative 12: 62.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

      \[\leadsto \left(\color{blue}{{x}^{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. sqr-pow N/A

      \[\leadsto \left(\color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   9. metadata-eval N/A

      \[\leadsto \left({x}^{\color{blue}{\frac{1}{4}}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   11. lower-pow.f64 N/A

       \[\leadsto \left({x}^{\frac{1}{4}} \cdot \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   12. metadata-eval 99.1

       \[\leadsto \left({x}^{0.25} \cdot \left({x}^{\color{blue}{0.25}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left({x}^{0.25} \cdot \left({x}^{0.25} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\frac{1}{4}} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left({x}^{\frac{1}{4}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto \left(\left(\color{blue}{{x}^{\frac{1}{4}}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   6. lift-pow.f64 N/A

      \[\leadsto \left(\left({x}^{\frac{1}{4}} \cdot \color{blue}{{x}^{\frac{1}{4}}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   7. pow-prod-up N/A

      \[\leadsto \left(\color{blue}{{x}^{\left(\frac{1}{4} + \frac{1}{4}\right)}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   8. metadata-eval N/A

      \[\leadsto \left({x}^{\color{blue}{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   9. pow1/2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate--l+ N/A

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

   15. distribute-rgt-in N/A

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

6. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 63.0%

    \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, \color{blue}{-3}\right) \]
12. Add Preprocessing

Alternative 13: 25.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ -3 \cdot \sqrt{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

      \[\leadsto \left(\color{blue}{{x}^{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. sqr-pow N/A

      \[\leadsto \left(\color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   9. metadata-eval N/A

      \[\leadsto \left({x}^{\color{blue}{\frac{1}{4}}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   11. lower-pow.f64 N/A

       \[\leadsto \left({x}^{\frac{1}{4}} \cdot \left(\color{blue}{{x}^{\left(\frac{\frac{1}{2}}{2}\right)}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   12. metadata-eval 99.1

       \[\leadsto \left({x}^{0.25} \cdot \left({x}^{\color{blue}{0.25}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left({x}^{0.25} \cdot \left({x}^{0.25} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot \left({x}^{\frac{1}{4}} \cdot 3\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left({x}^{\frac{1}{4}} \cdot \color{blue}{\left({x}^{\frac{1}{4}} \cdot 3\right)}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left({x}^{\frac{1}{4}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto \left(\left(\color{blue}{{x}^{\frac{1}{4}}} \cdot {x}^{\frac{1}{4}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   6. lift-pow.f64 N/A

      \[\leadsto \left(\left({x}^{\frac{1}{4}} \cdot \color{blue}{{x}^{\frac{1}{4}}}\right) \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   7. pow-prod-up N/A

      \[\leadsto \left(\color{blue}{{x}^{\left(\frac{1}{4} + \frac{1}{4}\right)}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   8. metadata-eval N/A

      \[\leadsto \left({x}^{\color{blue}{\frac{1}{2}}} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   9. pow1/2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate--l+ N/A

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

   15. distribute-rgt-in N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fma.f64 57.1

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

9. Applied rewrites 57.1%

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

10. Taylor expanded in y around 0

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

12. Applied rewrites 25.1%

    \[\leadsto -3 \cdot \color{blue}{\sqrt{x}} \]
13. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× 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 2024339 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x)))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))