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

Percentage Accurate: 99.4% → 99.4%

Time: 11.9s

Alternatives: 15

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0 - \frac{3}{\sqrt{x}}, \sqrt{x} \cdot \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma
  x
  (- 0.0 (/ 3.0 (sqrt x)))
  (* (sqrt x) (fma 3.0 y (/ 0.3333333333333333 x)))))

C

double code(double x, double y) {
	return fma(x, (0.0 - (3.0 / sqrt(x))), (sqrt(x) * fma(3.0, y, (0.3333333333333333 / x))));
}

Julia

function code(x, y)
	return fma(x, Float64(0.0 - Float64(3.0 / sqrt(x))), Float64(sqrt(x) * fma(3.0, y, Float64(0.3333333333333333 / x))))
end

Wolfram

code[x_, y_] := N[(x * N[(0.0 - N[(3.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 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. Step-by-step derivation

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

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

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

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

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

   7. clear-num N/A

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

   8. flip-- N/A

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

   9. /-lowering-/.f64 N/A

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

   10. associate--l+ N/A

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

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

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

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

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

4. Applied egg-rr 99.3%

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

   1. clear-num N/A

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

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

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

   3. associate-/l/ N/A

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

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

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

   5. /-rgt-identity N/A

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

   6. associate-/r/ N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

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

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

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

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

   11. div-inv N/A

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

   12. remove-double-div N/A

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

   13. associate-+r+ N/A

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

   14. distribute-lft-in N/A

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

   15. metadata-eval N/A

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

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

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

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

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

   18. /-lowering-/.f64 99.3

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

6. Applied egg-rr 99.3%

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

   1. remove-double-div N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-neg-in N/A

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

   7. remove-double-div N/A

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

   8. pow1/2 N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. pow-prod-up N/A

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

   13. pow1/2 N/A

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

   14. inv-pow N/A

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

   15. div-inv N/A

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

   16. clear-num N/A

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

   17. associate-/l* N/A

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

   18. *-commutative N/A

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

   19. associate-/l* N/A

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

   20. distribute-rgt-neg-in N/A

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(x, 0 - \frac{3}{\sqrt{x}}, \sqrt{x} \cdot \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\right) \]
10. Add Preprocessing

Alternative 2: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;t\_0 \cdot \left(y + -1\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x))) (t_1 (* t_0 (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_1 -1e+19)
     (* t_0 (+ y -1.0))
     (if (<= t_1 2e+153)
       (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
       (* (sqrt x) (fma 3.0 y -3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -1e+19) {
		tmp = t_0 * (y + -1.0);
	} else if (t_1 <= 2e+153) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_1 <= -1e+19)
		tmp = Float64(t_0 * Float64(y + -1.0));
	elseif (t_1 <= 2e+153)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+19], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+153], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $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))) < -1e19

   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

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

   5. Simplified 98.3%

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

   if -1e19 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 2e153

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. 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 82.8

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

   5. Simplified 82.8%

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

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

   1. Initial program 99.4%

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

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

   5. Simplified 99.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0 \cdot \left(y + -1\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x))) (t_1 (* t_0 (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_1 -1000.0)
     (* t_0 (+ y -1.0))
     (if (<= t_1 2e+153) (/ 1.0 t_0) (* (sqrt x) (fma 3.0 y -3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_0 * (y + -1.0);
	} else if (t_1 <= 2e+153) {
		tmp = 1.0 / t_0;
	} else {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(t_0 * Float64(y + -1.0));
	elseif (t_1 <= 2e+153)
		tmp = Float64(1.0 / t_0);
	else
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000.0], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+153], N[(1.0 / t$95$0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $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))) < -1e3

   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

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

   5. Simplified 97.6%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 2e153

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. 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 81.5

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

   5. Simplified 81.5%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 80.5

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

   8. Simplified 80.5%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lowering-*.f64 80.5

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

   10. Applied egg-rr 80.5%

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

       1. associate-/r* N/A

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

       2. div-inv N/A

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

       3. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{x}^{\frac{1}{2}}} \cdot \frac{1}{x}}{3} \]

       4. inv-pow N/A

          \[\leadsto \frac{{x}^{\frac{1}{2}} \cdot \color{blue}{{x}^{-1}}}{3} \]

       5. pow-prod-up N/A

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

       6. metadata-eval N/A

          \[\leadsto \frac{{x}^{\color{blue}{\frac{-1}{2}}}}{3} \]

       7. metadata-eval N/A

          \[\leadsto \frac{{x}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}}{3} \]

       8. pow-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{{x}^{\frac{1}{2}}}}}{3} \]

       9. pow1/2 N/A

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

       10. clear-num N/A

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

       11. div-inv N/A

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

       12. remove-double-div N/A

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

       13. *-commutative N/A

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

       14. /-lowering-/.f64 N/A

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

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

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

       16. sqrt-lowering-sqrt.f64 80.6

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

   12. Applied egg-rr 80.6%

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

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

   1. Initial program 99.4%

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

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

   5. Simplified 99.7%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -1000:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0 \cdot \left(y + -1\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x))) (t_1 (* t_0 (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_1 -1000.0)
     (* t_0 (+ y -1.0))
     (if (<= t_1 2e+153)
       (* (sqrt x) (/ 0.3333333333333333 x))
       (* (sqrt x) (fma 3.0 y -3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_0 * (y + -1.0);
	} else if (t_1 <= 2e+153) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(t_0 * Float64(y + -1.0));
	elseif (t_1 <= 2e+153)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	else
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000.0], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+153], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $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))) < -1e3

   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

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

   5. Simplified 97.6%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 2e153

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. 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 81.5

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

   5. Simplified 81.5%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 80.5

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

   8. Simplified 80.5%

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

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

   1. Initial program 99.4%

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

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

   5. Simplified 99.7%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -1000:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0 \cdot \left(y + -1\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x))) (t_1 (* t_0 (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_1 -1000.0)
     (* t_0 (+ y -1.0))
     (if (<= t_1 2e+153)
       (* 0.3333333333333333 (/ 1.0 (sqrt x)))
       (* (sqrt x) (fma 3.0 y -3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_0 * (y + -1.0);
	} else if (t_1 <= 2e+153) {
		tmp = 0.3333333333333333 * (1.0 / sqrt(x));
	} else {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(t_0 * Float64(y + -1.0));
	elseif (t_1 <= 2e+153)
		tmp = Float64(0.3333333333333333 * Float64(1.0 / sqrt(x)));
	else
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000.0], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+153], N[(0.3333333333333333 * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $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))) < -1e3

   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

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

   5. Simplified 97.6%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 2e153

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. 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 81.5

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

   5. Simplified 81.5%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 80.5

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

   8. Simplified 80.5%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow1/2 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot {x}^{\color{blue}{\frac{-1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot {x}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      9. pow-flip N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{1}{{x}^{\frac{1}{2}}}} \]

      10. pow1/2 N/A

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. sqrt-lowering-sqrt.f64 80.5

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

   10. Applied egg-rr 80.5%

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

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

   1. Initial program 99.4%

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

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

   5. Simplified 99.7%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -1000:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0 \cdot \left(y + -1\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x))) (t_1 (* t_0 (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_1 -1000.0)
     (* t_0 (+ y -1.0))
     (if (<= t_1 2e+153)
       (* 0.3333333333333333 (sqrt (/ 1.0 x)))
       (* (sqrt x) (fma 3.0 y -3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_0 * (y + -1.0);
	} else if (t_1 <= 2e+153) {
		tmp = 0.3333333333333333 * sqrt((1.0 / x));
	} else {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(t_0 * Float64(y + -1.0));
	elseif (t_1 <= 2e+153)
		tmp = Float64(0.3333333333333333 * sqrt(Float64(1.0 / x)));
	else
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000.0], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+153], N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $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))) < -1e3

   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

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

   5. Simplified 97.6%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 2e153

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. *-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 80.5

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

   5. Simplified 80.5%

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

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

   1. Initial program 99.4%

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

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

   5. Simplified 99.7%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -1000:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 2 \cdot 10^{+153}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ \mathbf{if}\;t\_0 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -1000:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x))))
   (if (<= (* t_0 (+ (+ y (/ 1.0 (* x 9.0))) -1.0)) -1000.0)
     (* (sqrt x) -3.0)
     t_0)))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double tmp;
	if ((t_0 * ((y + (1.0 / (x * 9.0))) + -1.0)) <= -1000.0) {
		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 = 3.0d0 * sqrt(x)
    if ((t_0 * ((y + (1.0d0 / (x * 9.0d0))) + (-1.0d0))) <= (-1000.0d0)) 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 = 3.0 * Math.sqrt(x);
	double tmp;
	if ((t_0 * ((y + (1.0 / (x * 9.0))) + -1.0)) <= -1000.0) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * math.sqrt(x)
	tmp = 0
	if (t_0 * ((y + (1.0 / (x * 9.0))) + -1.0)) <= -1000.0:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	tmp = 0.0
	if (Float64(t_0 * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0)) <= -1000.0)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * sqrt(x);
	tmp = 0.0;
	if ((t_0 * ((y + (1.0 / (x * 9.0))) + -1.0)) <= -1000.0)
		tmp = sqrt(x) * -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -1000.0], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 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))) < -1e3

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-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 53.6

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

   5. Simplified 53.6%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 53.4%

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

   if -1e3 < (*.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

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{9} \cdot \frac{1}{x} - 1\right)\right)} \]
   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 62.2

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

   5. Simplified 62.2%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. cube-mult N/A

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

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

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

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

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

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

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

      15. sqrt-lowering-sqrt.f64 62.2

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

   8. Simplified 62.2%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-3 \cdot \left(\sqrt{x} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-3 \cdot \sqrt{x}\right) \cdot {\left(\sqrt{-1}\right)}^{2}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{x} \cdot -3\right)} \cdot {\left(\sqrt{-1}\right)}^{2} \]

       3. unpow2 N/A

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

       4. rem-square-sqrt N/A

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

       5. associate-*l* N/A

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

       6. metadata-eval N/A

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

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

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

       8. sqrt-lowering-sqrt.f64 5.2

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

   11. Simplified 5.2%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -1000:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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]

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 9: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 15:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 15.0)
   (* (sqrt x) (fma 3.0 y (/ 0.3333333333333333 x)))
   (* (sqrt x) (fma 3.0 y -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 15.0) {
		tmp = sqrt(x) * fma(3.0, y, (0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 15.0)
		tmp = Float64(sqrt(x) * fma(3.0, y, Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 15.0], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 15

   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. Step-by-step derivation

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. associate--l+ N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. metadata-eval 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.0

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

   7. Simplified 98.0%

      \[\leadsto \left(\sqrt{x} \cdot \left(y + \color{blue}{\frac{0.1111111111111111}{x}}\right)\right) \cdot 3 \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

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

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. sqrt-lowering-sqrt.f64 98.2

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

   9. Applied egg-rr 98.2%

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

   if 15 < x

   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 98.9

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

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 15:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right) + -3, 0\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma (sqrt x) (+ (fma 3.0 y (/ 0.3333333333333333 x)) -3.0) 0.0))

C

double code(double x, double y) {
	return fma(sqrt(x), (fma(3.0, y, (0.3333333333333333 / x)) + -3.0), 0.0);
}

Julia

function code(x, y)
	return fma(sqrt(x), Float64(fma(3.0, y, Float64(0.3333333333333333 / x)) + -3.0), 0.0)
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision] + 0.0), $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

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

5. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right) + -3, 0\right)} \]
6. Add Preprocessing

Alternative 11: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

   4. *-commutative N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. distribute-lft-in N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

   11. +-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. /-lowering-/.f64 N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. sqrt-lowering-sqrt.f64 99.4

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

4. Applied egg-rr 99.4%

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

5. Final simplification 99.4%

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

Alternative 12: 60.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1350000000:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1350000000.0)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 1.0) (* (sqrt x) -3.0) (* (sqrt x) (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1350000000.0) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 1.0) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1350000000.0d0)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 1.0d0) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1350000000.0) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 1.0) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1350000000.0:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 1.0:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1350000000.0)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 1.0)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1350000000.0)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 1.0)
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1350000000.0], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35e9

   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. *-lowering-*.f64 N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 77.3

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

   5. Simplified 77.3%

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

   if -1.35e9 < y < 1

   1. Initial program 99.4%

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

   3. 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 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 52.5%

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

   if 1 < y

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 74.9

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

   5. Simplified 74.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 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. sqrt-lowering-sqrt.f64 75.0

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

   7. Applied egg-rr 75.0%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1350000000:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{if}\;y \leq -1350000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (* (sqrt x) y))))
   (if (<= y -1350000000.0) t_0 (if (<= y 1.0) (* (sqrt x) -3.0) t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (sqrt(x) * y);
	double tmp;
	if (y <= -1350000000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		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 = 3.0d0 * (sqrt(x) * y)
    if (y <= (-1350000000.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) 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 = 3.0 * (Math.sqrt(x) * y);
	double tmp;
	if (y <= -1350000000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * (math.sqrt(x) * y)
	tmp = 0
	if y <= -1350000000.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(sqrt(x) * y))
	tmp = 0.0
	if (y <= -1350000000.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * (sqrt(x) * y);
	tmp = 0.0;
	if (y <= -1350000000.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = sqrt(x) * -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1350000000.0], t$95$0, If[LessEqual[y, 1.0], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e9 or 1 < y

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 76.1

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

   5. Simplified 76.1%

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

   if -1.35e9 < y < 1

   1. Initial program 99.4%

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

   3. 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 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 52.5%

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

Alternative 14: 61.8% 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 66.0

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

5. Simplified 66.0%

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

Alternative 15: 25.6% 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

   \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{9} \cdot \frac{1}{x} - 1\right)\right)} \]
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 58.1

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

5. Simplified 58.1%

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

6. Taylor expanded in x around inf

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

8. Simplified 25.7%

   \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]
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 2024195 
(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)))