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

Percentage Accurate: 99.4% → 99.4%

Time: 6.6s

Alternatives: 12

Speedup: 1.2×

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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(3.0 * N[(N[(0.1111111111111111 / x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision] + N[(N[(y * N[Sqrt[x], $MachinePrecision]), $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. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(3 \cdot \left(\frac{0.1111111111111111}{x} - 1\right), \sqrt{x}, \left(y \cdot \sqrt{x}\right) \cdot 3\right) \]
6. Add Preprocessing

Alternative 2: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -70000000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) t_0)))
   (if (<= t_1 -70000000.0)
     (* (* (- y 1.0) 3.0) (sqrt x))
     (if (<= t_1 5e+152)
       (* (* (sqrt (/ 1.0 x)) (- 0.1111111111111111 x)) 3.0)
       (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -70000000.0) {
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	} else if (t_1 <= 5e+152) {
		tmp = (sqrt((1.0 / x)) * (0.1111111111111111 - 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 = sqrt(x) * 3.0d0
    t_1 = (((1.0d0 / (9.0d0 * x)) + y) - 1.0d0) * t_0
    if (t_1 <= (-70000000.0d0)) then
        tmp = ((y - 1.0d0) * 3.0d0) * sqrt(x)
    else if (t_1 <= 5d+152) then
        tmp = (sqrt((1.0d0 / x)) * (0.1111111111111111d0 - 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 = Math.sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -70000000.0) {
		tmp = ((y - 1.0) * 3.0) * Math.sqrt(x);
	} else if (t_1 <= 5e+152) {
		tmp = (Math.sqrt((1.0 / x)) * (0.1111111111111111 - x)) * 3.0;
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * 3.0
	t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0
	tmp = 0
	if t_1 <= -70000000.0:
		tmp = ((y - 1.0) * 3.0) * math.sqrt(x)
	elif t_1 <= 5e+152:
		tmp = (math.sqrt((1.0 / x)) * (0.1111111111111111 - x)) * 3.0
	else:
		tmp = t_0 * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * t_0)
	tmp = 0.0
	if (t_1 <= -70000000.0)
		tmp = Float64(Float64(Float64(y - 1.0) * 3.0) * sqrt(x));
	elseif (t_1 <= 5e+152)
		tmp = Float64(Float64(sqrt(Float64(1.0 / x)) * Float64(0.1111111111111111 - x)) * 3.0);
	else
		tmp = Float64(t_0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * 3.0;
	t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	tmp = 0.0;
	if (t_1 <= -70000000.0)
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	elseif (t_1 <= 5e+152)
		tmp = (sqrt((1.0 / x)) * (0.1111111111111111 - x)) * 3.0;
	else
		tmp = t_0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -70000000.0], N[(N[(N[(y - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+152], N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(0.1111111111111111 - x), $MachinePrecision]), $MachinePrecision] * 3.0), $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))) < -7e7

   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

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-sqrt.f64 N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

      12. lift--.f64 N/A

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

      13. associate--l+ N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-sqrt.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around inf

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

      1. lower--.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if -7e7 < (*.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))) < 5e152

   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. 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

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.4

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 99.4

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. lift-/.f64 N/A

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

      8. flip-+ N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. frac-add N/A

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

      12. lower-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower--.f64 86.9

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

   6. Applied rewrites 86.9%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 84.8

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

   9. Applied rewrites 84.8%

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

   if 5e152 < (*.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 99.7%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -70000000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -70000000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, 0.3333333333333333\right)}{x} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) t_0)))
   (if (<= t_1 -70000000.0)
     (* (* (- y 1.0) 3.0) (sqrt x))
     (if (<= t_1 5e+152)
       (* (/ (fma -3.0 x 0.3333333333333333) x) (sqrt x))
       (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -70000000.0) {
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	} else if (t_1 <= 5e+152) {
		tmp = (fma(-3.0, x, 0.3333333333333333) / x) * sqrt(x);
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * t_0)
	tmp = 0.0
	if (t_1 <= -70000000.0)
		tmp = Float64(Float64(Float64(y - 1.0) * 3.0) * sqrt(x));
	elseif (t_1 <= 5e+152)
		tmp = Float64(Float64(fma(-3.0, x, 0.3333333333333333) / x) * sqrt(x));
	else
		tmp = Float64(t_0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -70000000.0], N[(N[(N[(y - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+152], N[(N[(N[(-3.0 * x + 0.3333333333333333), $MachinePrecision] / x), $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))) < -7e7

   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

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-sqrt.f64 N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

      12. lift--.f64 N/A

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

      13. associate--l+ N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-sqrt.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around inf

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

      1. lower--.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if -7e7 < (*.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))) < 5e152

   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. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-sqrt.f64 N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

      12. lift--.f64 N/A

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

      13. associate--l+ N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-sqrt.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 99.4

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

   9. Applied rewrites 99.4%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 84.8%

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

   if 5e152 < (*.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 99.7%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -70000000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3, x, 0.3333333333333333\right)}{x} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -20:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) t_0)))
   (if (<= t_1 -20.0)
     (* (* (- y 1.0) 3.0) (sqrt x))
     (if (<= t_1 5e+152) (* (sqrt (/ 1.0 x)) 0.3333333333333333) (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -20.0) {
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	} else if (t_1 <= 5e+152) {
		tmp = sqrt((1.0 / x)) * 0.3333333333333333;
	} 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 = sqrt(x) * 3.0d0
    t_1 = (((1.0d0 / (9.0d0 * x)) + y) - 1.0d0) * t_0
    if (t_1 <= (-20.0d0)) then
        tmp = ((y - 1.0d0) * 3.0d0) * sqrt(x)
    else if (t_1 <= 5d+152) then
        tmp = sqrt((1.0d0 / x)) * 0.3333333333333333d0
    else
        tmp = t_0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -20.0) {
		tmp = ((y - 1.0) * 3.0) * Math.sqrt(x);
	} else if (t_1 <= 5e+152) {
		tmp = Math.sqrt((1.0 / x)) * 0.3333333333333333;
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * 3.0
	t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0
	tmp = 0
	if t_1 <= -20.0:
		tmp = ((y - 1.0) * 3.0) * math.sqrt(x)
	elif t_1 <= 5e+152:
		tmp = math.sqrt((1.0 / x)) * 0.3333333333333333
	else:
		tmp = t_0 * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * t_0)
	tmp = 0.0
	if (t_1 <= -20.0)
		tmp = Float64(Float64(Float64(y - 1.0) * 3.0) * sqrt(x));
	elseif (t_1 <= 5e+152)
		tmp = Float64(sqrt(Float64(1.0 / x)) * 0.3333333333333333);
	else
		tmp = Float64(t_0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * 3.0;
	t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	tmp = 0.0;
	if (t_1 <= -20.0)
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	elseif (t_1 <= 5e+152)
		tmp = sqrt((1.0 / x)) * 0.3333333333333333;
	else
		tmp = t_0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -20.0], N[(N[(N[(y - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+152], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $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))) < -20

   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

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-sqrt.f64 N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

      12. lift--.f64 N/A

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

      13. associate--l+ N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-sqrt.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around inf

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

      1. lower--.f64 98.2

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

   9. Applied rewrites 98.2%

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

   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))) < 5e152

   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

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt{\frac{1}{x}}} \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

   if 5e152 < (*.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 99.7%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -20:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(N[(y + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * 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. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-sqrt.f64 N/A

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

   10. distribute-rgt-out N/A

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

   11. +-commutative N/A

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

   12. lift--.f64 N/A

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

   13. associate--l+ N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-sqrt.f64 N/A

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

6. Applied rewrites 99.5%

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

Alternative 6: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-in N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

Alternative 7: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - 1, 3, \frac{0.3333333333333333}{x}\right) \cdot \sqrt{x} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (fma (- y 1.0) 3.0 (/ 0.3333333333333333 x)) (sqrt x)))

C

double code(double x, double y) {
	return fma((y - 1.0), 3.0, (0.3333333333333333 / x)) * sqrt(x);
}

Julia

function code(x, y)
	return Float64(fma(Float64(y - 1.0), 3.0, Float64(0.3333333333333333 / x)) * sqrt(x))
end

Wolfram

code[x_, y_] := N[(N[(N[(y - 1.0), $MachinePrecision] * 3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] * 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. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-sqrt.f64 N/A

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

   10. distribute-rgt-out N/A

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

   11. +-commutative N/A

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

   12. lift--.f64 N/A

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

   13. associate--l+ N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-sqrt.f64 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 99.4

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

9. Applied rewrites 99.4%

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

Alternative 8: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -650:\\ \;\;\;\;\left(y \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 0.19:\\ \;\;\;\;\left(-\sqrt{x}\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -650.0)
   (* (* y 3.0) (sqrt x))
   (if (<= y 0.19) (* (- (sqrt x)) 3.0) (* (* (sqrt x) 3.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -650.0) {
		tmp = (y * 3.0) * sqrt(x);
	} else if (y <= 0.19) {
		tmp = -sqrt(x) * 3.0;
	} else {
		tmp = (sqrt(x) * 3.0) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-650.0d0)) then
        tmp = (y * 3.0d0) * sqrt(x)
    else if (y <= 0.19d0) then
        tmp = -sqrt(x) * 3.0d0
    else
        tmp = (sqrt(x) * 3.0d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -650.0) {
		tmp = (y * 3.0) * Math.sqrt(x);
	} else if (y <= 0.19) {
		tmp = -Math.sqrt(x) * 3.0;
	} else {
		tmp = (Math.sqrt(x) * 3.0) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -650.0:
		tmp = (y * 3.0) * math.sqrt(x)
	elif y <= 0.19:
		tmp = -math.sqrt(x) * 3.0
	else:
		tmp = (math.sqrt(x) * 3.0) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -650.0)
		tmp = Float64(Float64(y * 3.0) * sqrt(x));
	elseif (y <= 0.19)
		tmp = Float64(Float64(-sqrt(x)) * 3.0);
	else
		tmp = Float64(Float64(sqrt(x) * 3.0) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -650.0)
		tmp = (y * 3.0) * sqrt(x);
	elseif (y <= 0.19)
		tmp = -sqrt(x) * 3.0;
	else
		tmp = (sqrt(x) * 3.0) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -650.0], N[(N[(y * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.19], N[((-N[Sqrt[x], $MachinePrecision]) * 3.0), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -650

   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 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 82.1%

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

   if -650 < y < 0.19

   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

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.4

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 99.4

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

   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 y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 44.9%

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

   if 0.19 < 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 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 75.2%

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

Alternative 9: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -650:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.19:\\ \;\;\;\;\left(-\sqrt{x}\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 3.0) (sqrt x))))
   (if (<= y -650.0) t_0 (if (<= y 0.19) (* (- (sqrt x)) 3.0) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 3.0) * sqrt(x);
	double tmp;
	if (y <= -650.0) {
		tmp = t_0;
	} else if (y <= 0.19) {
		tmp = -sqrt(x) * 3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * 3.0d0) * sqrt(x)
    if (y <= (-650.0d0)) then
        tmp = t_0
    else if (y <= 0.19d0) then
        tmp = -sqrt(x) * 3.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 3.0) * Math.sqrt(x);
	double tmp;
	if (y <= -650.0) {
		tmp = t_0;
	} else if (y <= 0.19) {
		tmp = -Math.sqrt(x) * 3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * 3.0) * math.sqrt(x)
	tmp = 0
	if y <= -650.0:
		tmp = t_0
	elif y <= 0.19:
		tmp = -math.sqrt(x) * 3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 3.0) * sqrt(x))
	tmp = 0.0
	if (y <= -650.0)
		tmp = t_0;
	elseif (y <= 0.19)
		tmp = Float64(Float64(-sqrt(x)) * 3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * 3.0) * sqrt(x);
	tmp = 0.0;
	if (y <= -650.0)
		tmp = t_0;
	elseif (y <= 0.19)
		tmp = -sqrt(x) * 3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -650.0], t$95$0, If[LessEqual[y, 0.19], N[((-N[Sqrt[x], $MachinePrecision]) * 3.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -650 or 0.19 < 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 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 79.4%

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

   if -650 < y < 0.19

   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

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.4

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 99.4

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

   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 y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 44.9%

       \[\leadsto \left(-\sqrt{x}\right) \cdot 3 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(y - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * 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. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-sqrt.f64 N/A

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

   10. distribute-rgt-out N/A

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

   11. +-commutative N/A

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

   12. lift--.f64 N/A

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

   13. associate--l+ N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-sqrt.f64 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around inf

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

   1. lower--.f64 61.6

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

9. Applied rewrites 61.6%

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

Alternative 11: 62.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y - 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. 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. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower--.f64 61.5

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

5. Applied rewrites 61.5%

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

6. Final simplification 61.5%

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

Alternative 12: 25.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.5

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 99.5

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

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 y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-sqrt.f64 62.8

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

7. Applied rewrites 62.8%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 25.7%

    \[\leadsto \left(-\sqrt{x}\right) \cdot 3 \]
11. 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 2024298 
(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)))