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

Percentage Accurate: 99.4% → 99.4%

Time: 10.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(3.0 * y + N[(1.0 / N[(x * 3.0), $MachinePrecision]), $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. lower-*.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. lower-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. lower-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. lower-+.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. lower-/.f64 99.4

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 2: 92.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+153}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \mathsf{fma}\left(x, -3, 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_0 -2e+40)
     (* (sqrt x) (fma 3.0 y -3.0))
     (if (<= t_0 1e+153)
       (* (sqrt (/ 1.0 x)) (fma x -3.0 0.3333333333333333))
       (* 3.0 (* y (sqrt x)))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_0 <= -2e+40) {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	} else if (t_0 <= 1e+153) {
		tmp = sqrt((1.0 / x)) * fma(x, -3.0, 0.3333333333333333);
	} else {
		tmp = 3.0 * (y * sqrt(x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_0 <= -2e+40)
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	elseif (t_0 <= 1e+153)
		tmp = Float64(sqrt(Float64(1.0 / x)) * fma(x, -3.0, 0.3333333333333333));
	else
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -2e+40], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+153], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(x * -3.0 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(y * N[Sqrt[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))) < -2.00000000000000006e40

   1. Initial program 99.5%

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

   3. 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. lower-*.f64 N/A

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

      4. lower-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. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. cube-mult N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sqrt.f64 89.8

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

   5. Applied rewrites 89.8%

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

   7. Applied rewrites 96.2%

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

   9. Applied rewrites 96.2%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 84.7%

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

   if 1e153 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 97.4

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

   7. Applied rewrites 97.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 10^{+153}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \mathsf{fma}\left(x, -3, 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_0 -5e+73)
     (* (sqrt x) (fma 3.0 y -3.0))
     (if (<= t_0 1e+153)
       (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
       (* 3.0 (* y (sqrt x)))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_0 <= -5e+73) {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	} else if (t_0 <= 1e+153) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = 3.0 * (y * sqrt(x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_0 <= -5e+73)
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	elseif (t_0 <= 1e+153)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -5e+73], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+153], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(y * N[Sqrt[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))) < -4.99999999999999976e73

   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. lower-*.f64 N/A

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

      4. lower-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. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -4.99999999999999976e73 < (*.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))) < 1e153

   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

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{9} \cdot \frac{1}{x} - 1\right)\right)} \]
   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. lower-*.f64 N/A

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

      5. lower-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. lower-+.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. lower-/.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if 1e153 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 97.4

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

   7. Applied rewrites 97.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -5 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_0 \leq -0.4:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+153}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_0 -0.4)
     (* (sqrt x) (fma 3.0 y -3.0))
     (if (<= t_0 1e+153)
       (/ 0.3333333333333333 (sqrt x))
       (* 3.0 (* y (sqrt x)))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_0 <= -0.4) {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	} else if (t_0 <= 1e+153) {
		tmp = 0.3333333333333333 / sqrt(x);
	} else {
		tmp = 3.0 * (y * sqrt(x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_0 <= -0.4)
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	elseif (t_0 <= 1e+153)
		tmp = Float64(0.3333333333333333 / sqrt(x));
	else
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -0.4], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+153], N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(y * N[Sqrt[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))) < -0.40000000000000002

   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. lower-*.f64 N/A

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

      4. lower-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. lower-fma.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if -0.40000000000000002 < (*.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))) < 1e153

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.4

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 82.5

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

   7. Applied rewrites 82.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.6%

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

   if 1e153 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 97.4

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

   7. Applied rewrites 97.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -0.4:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 10^{+153}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(3.0 * y + 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. lower-*.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. lower-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. lower-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. lower-+.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. lower-/.f64 99.4

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.5%

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

Alternative 6: 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. lower-*.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. lower-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. lower-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. lower-+.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. lower-/.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 7: 62.1% 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. lower-*.f64 N/A

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

   4. lower-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. lower-fma.f64 62.7

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

5. Applied rewrites 62.7%

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

Alternative 8: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   6. lower-sqrt.f64 34.6

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

5. Applied rewrites 34.6%

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

7. Applied rewrites 34.7%

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

8. Final simplification 34.7%

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

Alternative 9: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   6. lower-sqrt.f64 34.6

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

5. Applied rewrites 34.6%

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

6. Final simplification 34.6%

   \[\leadsto y \cdot \left(3 \cdot \sqrt{x}\right) \]
7. 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 2024219 
(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)))