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

Percentage Accurate: 99.4% → 99.4%

Time: 9.8s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. *-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. distribute-rgt-in N/A

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

   15. metadata-eval N/A

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

   16. +-lowering-+.f64 N/A

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

   17. associate-*r/ N/A

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

   18. metadata-eval N/A

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

   19. associate-*l/ N/A

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

   20. metadata-eval N/A

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

   21. /-lowering-/.f64 99.5

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

5. Simplified 99.5%

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

Alternative 2: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;t\_0 \leq 1000000:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(\frac{3}{x}, 0.1111111111111111, -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (sqrt x) 3.0) (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_0 -2e+63)
     (* (sqrt x) (fma 3.0 y -3.0))
     (if (<= t_0 1000000.0)
       (* (sqrt x) (fma (/ 3.0 x) 0.1111111111111111 -3.0))
       (* (sqrt x) (fma 3.0 y (/ 0.3333333333333333 x)))))))

C

double code(double x, double y) {
	double t_0 = (sqrt(x) * 3.0) * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_0 <= -2e+63) {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	} else if (t_0 <= 1000000.0) {
		tmp = sqrt(x) * fma((3.0 / x), 0.1111111111111111, -3.0);
	} else {
		tmp = sqrt(x) * fma(3.0, y, (0.3333333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(x) * 3.0) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_0 <= -2e+63)
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	elseif (t_0 <= 1000000.0)
		tmp = Float64(sqrt(x) * fma(Float64(3.0 / x), 0.1111111111111111, -3.0));
	else
		tmp = Float64(sqrt(x) * fma(3.0, y, Float64(0.3333333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+63], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 / x), $MachinePrecision] * 0.1111111111111111 + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -2.00000000000000012e63

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.1

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

   5. Simplified 99.1%

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

   if -2.00000000000000012e63 < (*.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))) < 1e6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. associate-*l/ N/A

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

      20. metadata-eval N/A

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

      21. /-lowering-/.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 98.1

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

   8. Simplified 98.1%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 98.1

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

   10. Applied egg-rr 98.1%

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

   if 1e6 < (*.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 99.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. associate-*l/ N/A

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

      20. metadata-eval N/A

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

      21. /-lowering-/.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.4

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

   8. Simplified 99.4%

      \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, \color{blue}{\frac{0.3333333333333333}{x}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 1000000:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(\frac{3}{x}, 0.1111111111111111, -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;t\_0 \leq 1000000:\\ \;\;\;\;\sqrt{x} \cdot \frac{\mathsf{fma}\left(x, -3, 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (sqrt x) 3.0) (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_0 -1e+36)
     (* (sqrt x) (fma 3.0 y -3.0))
     (if (<= t_0 1000000.0)
       (* (sqrt x) (/ (fma x -3.0 0.3333333333333333) x))
       (* (sqrt x) (fma 3.0 y (/ 0.3333333333333333 x)))))))

C

double code(double x, double y) {
	double t_0 = (sqrt(x) * 3.0) * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_0 <= -1e+36) {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	} else if (t_0 <= 1000000.0) {
		tmp = sqrt(x) * (fma(x, -3.0, 0.3333333333333333) / x);
	} else {
		tmp = sqrt(x) * fma(3.0, y, (0.3333333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(x) * 3.0) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_0 <= -1e+36)
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	elseif (t_0 <= 1000000.0)
		tmp = Float64(sqrt(x) * Float64(fma(x, -3.0, 0.3333333333333333) / x));
	else
		tmp = Float64(sqrt(x) * fma(3.0, y, Float64(0.3333333333333333 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+36], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(x * -3.0 + 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -1.00000000000000004e36

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.1

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

   5. Simplified 99.1%

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

   if -1.00000000000000004e36 < (*.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))) < 1e6

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. associate-*l/ N/A

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

      20. metadata-eval N/A

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

      21. /-lowering-/.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 97.5

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

   8. Simplified 97.5%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. accelerator-lowering-fma.f64 97.5

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

   11. Simplified 97.5%

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

   if 1e6 < (*.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 99.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. associate-*l/ N/A

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

      20. metadata-eval N/A

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

      21. /-lowering-/.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.4

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

   8. Simplified 99.4%

      \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, \color{blue}{\frac{0.3333333333333333}{x}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 1000000:\\ \;\;\;\;\sqrt{x} \cdot \frac{\mathsf{fma}\left(x, -3, 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\mathsf{fma}\left(x, -3, 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (sqrt x) 3.0) (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_0 -1e+36)
     (* (sqrt x) (fma 3.0 y -3.0))
     (if (<= t_0 5e+152)
       (* (sqrt x) (/ (fma x -3.0 0.3333333333333333) x))
       (* (sqrt x) (* 3.0 y))))))

C

double code(double x, double y) {
	double t_0 = (sqrt(x) * 3.0) * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_0 <= -1e+36) {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	} else if (t_0 <= 5e+152) {
		tmp = sqrt(x) * (fma(x, -3.0, 0.3333333333333333) / x);
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(x) * 3.0) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_0 <= -1e+36)
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	elseif (t_0 <= 5e+152)
		tmp = Float64(sqrt(x) * Float64(fma(x, -3.0, 0.3333333333333333) / x));
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+36], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+152], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(x * -3.0 + 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -1.00000000000000004e36

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.1

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

   5. Simplified 99.1%

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

   if -1.00000000000000004e36 < (*.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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. associate-*l/ N/A

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

      20. metadata-eval N/A

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

      21. /-lowering-/.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 88.0

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

   8. Simplified 88.0%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. accelerator-lowering-fma.f64 88.0

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

   11. Simplified 88.0%

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

   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 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 99.5

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

   5. Simplified 99.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 99.7

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -1 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\mathsf{fma}\left(x, -3, 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (sqrt x) 3.0) (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_0 -2e+63)
     (* (sqrt x) (fma 3.0 y -3.0))
     (if (<= t_0 5e+152)
       (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
       (* (sqrt x) (* 3.0 y))))))

C

double code(double x, double y) {
	double t_0 = (sqrt(x) * 3.0) * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_0 <= -2e+63) {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	} else if (t_0 <= 5e+152) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(x) * 3.0) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_0 <= -2e+63)
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	elseif (t_0 <= 5e+152)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+63], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+152], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -2.00000000000000012e63

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 99.1

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

   5. Simplified 99.1%

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

   if -2.00000000000000012e63 < (*.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. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 89.0

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

   5. Simplified 89.0%

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

   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 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 99.5

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

   5. Simplified 99.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 99.7

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -2 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (sqrt x) 3.0) (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_0 -0.5)
     (* (sqrt x) (fma 3.0 y -3.0))
     (if (<= t_0 5e+152)
       (* 0.3333333333333333 (sqrt (/ 1.0 x)))
       (* (sqrt x) (* 3.0 y))))))

C

double code(double x, double y) {
	double t_0 = (sqrt(x) * 3.0) * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	} else if (t_0 <= 5e+152) {
		tmp = 0.3333333333333333 * sqrt((1.0 / x));
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(x) * 3.0) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	elseif (t_0 <= 5e+152)
		tmp = Float64(0.3333333333333333 * sqrt(Float64(1.0 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+152], N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -0.5

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 95.8

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

   5. Simplified 95.8%

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

   if -0.5 < (*.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 99.2%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 85.2

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

   5. Simplified 85.2%

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

   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 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 99.5

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

   5. Simplified 99.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 99.7

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -0.5:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{if}\;y \leq -45000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (* 3.0 y))))
   (if (<= y -45000.0) t_0 (if (<= y 5e-6) (* (sqrt x) -3.0) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * (3.0 * y);
	double tmp;
	if (y <= -45000.0) {
		tmp = t_0;
	} else if (y <= 5e-6) {
		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 = sqrt(x) * (3.0d0 * y)
    if (y <= (-45000.0d0)) then
        tmp = t_0
    else if (y <= 5d-6) 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 = Math.sqrt(x) * (3.0 * y);
	double tmp;
	if (y <= -45000.0) {
		tmp = t_0;
	} else if (y <= 5e-6) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * (3.0 * y)
	tmp = 0
	if y <= -45000.0:
		tmp = t_0
	elif y <= 5e-6:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * Float64(3.0 * y))
	tmp = 0.0
	if (y <= -45000.0)
		tmp = t_0;
	elseif (y <= 5e-6)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * (3.0 * y);
	tmp = 0.0;
	if (y <= -45000.0)
		tmp = t_0;
	elseif (y <= 5e-6)
		tmp = sqrt(x) * -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -45000.0], t$95$0, If[LessEqual[y, 5e-6], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -45000 or 5.00000000000000041e-6 < 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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 76.2

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

   5. Simplified 76.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 76.3

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

   7. Applied egg-rr 76.3%

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

   if -45000 < y < 5.00000000000000041e-6

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 47.4

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

   8. Simplified 47.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -45000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;y \leq -45000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* (sqrt x) 3.0))))
   (if (<= y -45000.0) t_0 (if (<= y 5e-6) (* (sqrt x) -3.0) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (sqrt(x) * 3.0);
	double tmp;
	if (y <= -45000.0) {
		tmp = t_0;
	} else if (y <= 5e-6) {
		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 * (sqrt(x) * 3.0d0)
    if (y <= (-45000.0d0)) then
        tmp = t_0
    else if (y <= 5d-6) 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 * (Math.sqrt(x) * 3.0);
	double tmp;
	if (y <= -45000.0) {
		tmp = t_0;
	} else if (y <= 5e-6) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (math.sqrt(x) * 3.0)
	tmp = 0
	if y <= -45000.0:
		tmp = t_0
	elif y <= 5e-6:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(sqrt(x) * 3.0))
	tmp = 0.0
	if (y <= -45000.0)
		tmp = t_0;
	elseif (y <= 5e-6)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (sqrt(x) * 3.0);
	tmp = 0.0;
	if (y <= -45000.0)
		tmp = t_0;
	elseif (y <= 5e-6)
		tmp = sqrt(x) * -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -45000.0], t$95$0, If[LessEqual[y, 5e-6], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -45000 or 5.00000000000000041e-6 < 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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 76.2

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

   5. Simplified 76.2%

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

   if -45000 < y < 5.00000000000000041e-6

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 47.4

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

   8. Simplified 47.4%

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

Alternative 9: 62.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. 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. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-in N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 63.2

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

5. Simplified 63.2%

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

Alternative 10: 26.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 59.0

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

5. Simplified 59.0%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 23.6

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

8. Simplified 23.6%

   \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
9. 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 2024199 
(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)))