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

Percentage Accurate: 99.4% → 99.4%

Time: 8.0s

Alternatives: 12

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 99.5%

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

3. Final simplification 99.5%

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

Alternative 2: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+32}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x)))
        (t_1 (* t_0 (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_1 -2e+32)
     (* (- y 1.0) (* (sqrt x) 3.0))
     (if (<= t_1 2e+151)
       (* (sqrt x) (- (/ 0.3333333333333333 x) 3.0))
       (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -2e+32) {
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	} else if (t_1 <= 2e+151) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) - 3.0);
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 * sqrt(x)
    t_1 = t_0 * ((y + ((x * 9.0d0) ** (-1.0d0))) - 1.0d0)
    if (t_1 <= (-2d+32)) then
        tmp = (y - 1.0d0) * (sqrt(x) * 3.0d0)
    else if (t_1 <= 2d+151) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) - 3.0d0)
    else
        tmp = t_0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * Math.sqrt(x);
	double t_1 = t_0 * ((y + Math.pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -2e+32) {
		tmp = (y - 1.0) * (Math.sqrt(x) * 3.0);
	} else if (t_1 <= 2e+151) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) - 3.0);
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * math.sqrt(x)
	t_1 = t_0 * ((y + math.pow((x * 9.0), -1.0)) - 1.0)
	tmp = 0
	if t_1 <= -2e+32:
		tmp = (y - 1.0) * (math.sqrt(x) * 3.0)
	elif t_1 <= 2e+151:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) - 3.0)
	else:
		tmp = t_0 * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_1 <= -2e+32)
		tmp = Float64(Float64(y - 1.0) * Float64(sqrt(x) * 3.0));
	elseif (t_1 <= 2e+151)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) - 3.0));
	else
		tmp = Float64(t_0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * sqrt(x);
	t_1 = t_0 * ((y + ((x * 9.0) ^ -1.0)) - 1.0);
	tmp = 0.0;
	if (t_1 <= -2e+32)
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	elseif (t_1 <= 2e+151)
		tmp = sqrt(x) * ((0.3333333333333333 / x) - 3.0);
	else
		tmp = t_0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+32], N[(N[(y - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+151], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * y), $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))) < -2.00000000000000011e32

   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. *-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 99.5

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   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 98.5

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

   7. Applied rewrites 98.5%

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

   if -2.00000000000000011e32 < (*.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))) < 2.00000000000000003e151

   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. 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. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-out-- N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 83.5%

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

   if 2.00000000000000003e151 < (*.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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -2 \cdot 10^{+32}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, 0.3333333333333333\right)}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x)))
        (t_1 (* t_0 (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_1 -5e+21)
     (* (- y 1.0) (* (sqrt x) 3.0))
     (if (<= t_1 2e+151)
       (/ (fma -3.0 x 0.3333333333333333) (sqrt x))
       (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -5e+21) {
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	} else if (t_1 <= 2e+151) {
		tmp = fma(-3.0, x, 0.3333333333333333) / sqrt(x);
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_1 <= -5e+21)
		tmp = Float64(Float64(y - 1.0) * Float64(sqrt(x) * 3.0));
	elseif (t_1 <= 2e+151)
		tmp = Float64(fma(-3.0, x, 0.3333333333333333) / sqrt(x));
	else
		tmp = Float64(t_0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+21], N[(N[(y - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+151], N[(N[(-3.0 * x + 0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * y), $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))) < -5e21

   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. *-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 99.5

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   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 98.5

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

   7. Applied rewrites 98.5%

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

   if -5e21 < (*.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))) < 2.00000000000000003e151

   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 0

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 87.5%

      \[\leadsto \sqrt{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(3, y, 3\right), x \cdot \mathsf{fma}\left(y \cdot y, 9, -9\right)\right)}{\color{blue}{x \cdot \mathsf{fma}\left(3, y, 3\right)}} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 83.1%

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

   12. Applied rewrites 83.0%

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

   if 2.00000000000000003e151 < (*.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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, 0.3333333333333333\right)}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x)))
        (t_1 (* t_0 (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_1 -200.0)
     (* (- y 1.0) (* (sqrt x) 3.0))
     (if (<= t_1 2e+151) (/ (* 0.3333333333333333 (sqrt x)) x) (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -200.0) {
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	} else if (t_1 <= 2e+151) {
		tmp = (0.3333333333333333 * sqrt(x)) / x;
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 * sqrt(x)
    t_1 = t_0 * ((y + ((x * 9.0d0) ** (-1.0d0))) - 1.0d0)
    if (t_1 <= (-200.0d0)) then
        tmp = (y - 1.0d0) * (sqrt(x) * 3.0d0)
    else if (t_1 <= 2d+151) then
        tmp = (0.3333333333333333d0 * sqrt(x)) / x
    else
        tmp = t_0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * Math.sqrt(x);
	double t_1 = t_0 * ((y + Math.pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -200.0) {
		tmp = (y - 1.0) * (Math.sqrt(x) * 3.0);
	} else if (t_1 <= 2e+151) {
		tmp = (0.3333333333333333 * Math.sqrt(x)) / x;
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * math.sqrt(x)
	t_1 = t_0 * ((y + math.pow((x * 9.0), -1.0)) - 1.0)
	tmp = 0
	if t_1 <= -200.0:
		tmp = (y - 1.0) * (math.sqrt(x) * 3.0)
	elif t_1 <= 2e+151:
		tmp = (0.3333333333333333 * math.sqrt(x)) / x
	else:
		tmp = t_0 * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = Float64(Float64(y - 1.0) * Float64(sqrt(x) * 3.0));
	elseif (t_1 <= 2e+151)
		tmp = Float64(Float64(0.3333333333333333 * sqrt(x)) / x);
	else
		tmp = Float64(t_0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * sqrt(x);
	t_1 = t_0 * ((y + ((x * 9.0) ^ -1.0)) - 1.0);
	tmp = 0.0;
	if (t_1 <= -200.0)
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	elseif (t_1 <= 2e+151)
		tmp = (0.3333333333333333 * sqrt(x)) / x;
	else
		tmp = t_0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], N[(N[(y - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+151], N[(N[(0.3333333333333333 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(t$95$0 * y), $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))) < -200

   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. 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 99.6

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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 97.7

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

   7. Applied rewrites 97.7%

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

   if -200 < (*.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))) < 2.00000000000000003e151

   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 x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \sqrt{x} + 3 \cdot \left(\sqrt{{x}^{3}} \cdot \left(y - 1\right)\right)}{x}} \]
   4. 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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sqrt.f64 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

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

   if 2.00000000000000003e151 < (*.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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -200:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x)))
        (t_1 (* t_0 (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_1 -200.0)
     (* (- y 1.0) (* (sqrt x) 3.0))
     (if (<= t_1 2e+151) (* (sqrt x) (/ 0.3333333333333333 x)) (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -200.0) {
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	} else if (t_1 <= 2e+151) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 * sqrt(x)
    t_1 = t_0 * ((y + ((x * 9.0d0) ** (-1.0d0))) - 1.0d0)
    if (t_1 <= (-200.0d0)) then
        tmp = (y - 1.0d0) * (sqrt(x) * 3.0d0)
    else if (t_1 <= 2d+151) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else
        tmp = t_0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * Math.sqrt(x);
	double t_1 = t_0 * ((y + Math.pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -200.0) {
		tmp = (y - 1.0) * (Math.sqrt(x) * 3.0);
	} else if (t_1 <= 2e+151) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * math.sqrt(x)
	t_1 = t_0 * ((y + math.pow((x * 9.0), -1.0)) - 1.0)
	tmp = 0
	if t_1 <= -200.0:
		tmp = (y - 1.0) * (math.sqrt(x) * 3.0)
	elif t_1 <= 2e+151:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	else:
		tmp = t_0 * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = Float64(Float64(y - 1.0) * Float64(sqrt(x) * 3.0));
	elseif (t_1 <= 2e+151)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	else
		tmp = Float64(t_0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * sqrt(x);
	t_1 = t_0 * ((y + ((x * 9.0) ^ -1.0)) - 1.0);
	tmp = 0.0;
	if (t_1 <= -200.0)
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	elseif (t_1 <= 2e+151)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	else
		tmp = t_0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], N[(N[(y - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+151], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * y), $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))) < -200

   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. 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 99.6

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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 97.7

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

   7. Applied rewrites 97.7%

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

   if -200 < (*.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))) < 2.00000000000000003e151

   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 0

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

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

   if 2.00000000000000003e151 < (*.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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -200:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 99.5%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
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. *-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 99.5

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.5

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 7: 99.4% accurate, 1.2× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

5. Applied rewrites 99.4%

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

Alternative 8: 60.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15500 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -15500.0) (not (<= y 1.0)))
   (* (* 3.0 (sqrt x)) y)
   (* -3.0 (sqrt x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -15500.0) || !(y <= 1.0)) {
		tmp = (3.0 * sqrt(x)) * y;
	} 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 <= (-15500.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = (3.0d0 * sqrt(x)) * y
    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 <= -15500.0) || !(y <= 1.0)) {
		tmp = (3.0 * Math.sqrt(x)) * y;
	} else {
		tmp = -3.0 * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -15500.0) or not (y <= 1.0):
		tmp = (3.0 * math.sqrt(x)) * y
	else:
		tmp = -3.0 * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -15500.0) || !(y <= 1.0))
		tmp = Float64(Float64(3.0 * sqrt(x)) * y);
	else
		tmp = Float64(-3.0 * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -15500.0) || ~((y <= 1.0)))
		tmp = (3.0 * sqrt(x)) * y;
	else
		tmp = -3.0 * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -15500.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -15500 or 1 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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 72.6

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

   5. Applied rewrites 72.6%

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

   7. Applied rewrites 72.7%

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

   if -15500 < y < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.1%

      \[\leadsto \sqrt{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(3, y, 3\right), x \cdot \mathsf{fma}\left(y \cdot y, 9, -9\right)\right)}{\color{blue}{x \cdot \mathsf{fma}\left(3, y, 3\right)}} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 45.9%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15500 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15500 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -15500.0) (not (<= y 1.0)))
   (* (* 3.0 y) (sqrt x))
   (* -3.0 (sqrt x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -15500.0) || !(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 <= (-15500.0d0)) .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 <= -15500.0) || !(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 <= -15500.0) 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 <= -15500.0) || !(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 <= -15500.0) || ~((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, -15500.0], 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 < -15500 or 1 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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 72.6

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

   5. Applied rewrites 72.6%

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

   7. Applied rewrites 72.5%

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

   if -15500 < y < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.1%

      \[\leadsto \sqrt{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(3, y, 3\right), x \cdot \mathsf{fma}\left(y \cdot y, 9, -9\right)\right)}{\color{blue}{x \cdot \mathsf{fma}\left(3, y, 3\right)}} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 45.9%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15500 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.5% 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 99.5%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
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. *-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 99.5

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.5

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

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 60.0

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

7. Applied rewrites 60.0%

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

Alternative 11: 61.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-sqrt.f64 60.0

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

5. Applied rewrites 60.0%

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

Alternative 12: 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 99.5%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 68.4%

   \[\leadsto \sqrt{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(3, y, 3\right), x \cdot \mathsf{fma}\left(y \cdot y, 9, -9\right)\right)}{\color{blue}{x \cdot \mathsf{fma}\left(3, y, 3\right)}} \]

8. Taylor expanded in y around 0

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

10. Applied rewrites 62.0%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 24.3%

    \[\leadsto -3 \cdot \sqrt{x} \]
14. 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 2024318 
(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)))