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

Percentage Accurate: 99.4% → 99.4%

Time: 10.5s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. 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 99.4

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

4. Applied egg-rr 99.4%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   9. sqrt-lowering-sqrt.f64 99.4

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

6. Applied egg-rr 99.4%

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

Alternative 2: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+34}:\\ \;\;\;\;3 \cdot \mathsf{fma}\left(\sqrt{x}, y, -\sqrt{x}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (+ -1.0 (+ y (/ 1.0 (* x 9.0)))) t_0)))
   (if (<= t_1 -2e+34)
     (* 3.0 (fma (sqrt x) y (- (sqrt x))))
     (if (<= t_1 1e+153)
       (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
       (* y t_0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	double tmp;
	if (t_1 <= -2e+34) {
		tmp = 3.0 * fma(sqrt(x), y, -sqrt(x));
	} else if (t_1 <= 1e+153) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(-1.0 + Float64(y + Float64(1.0 / Float64(x * 9.0)))) * t_0)
	tmp = 0.0
	if (t_1 <= -2e+34)
		tmp = Float64(3.0 * fma(sqrt(x), y, Float64(-sqrt(x))));
	elseif (t_1 <= 1e+153)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(y * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 + N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+34], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+153], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -1.99999999999999989e34

   1. Initial program 99.6%

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

      1. 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 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 99.6

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

   7. Simplified 99.6%

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

      1. distribute-lft-in N/A

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

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

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

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

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 99.6

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

   9. Applied egg-rr 99.6%

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

   if -1.99999999999999989e34 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 1e153

   1. Initial program 99.2%

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

   3. Taylor expanded in 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 86.7

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

   5. Simplified 86.7%

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

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

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 95.4

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

   5. Simplified 95.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -2 \cdot 10^{+34}:\\ \;\;\;\;3 \cdot \mathsf{fma}\left(\sqrt{x}, y, -\sqrt{x}\right)\\ \mathbf{elif}\;\left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+34}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(-1 + y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (+ -1.0 (+ y (/ 1.0 (* x 9.0)))) t_0)))
   (if (<= t_1 -2e+34)
     (* 3.0 (* (sqrt x) (+ -1.0 y)))
     (if (<= t_1 1e+153)
       (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
       (* y t_0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	double tmp;
	if (t_1 <= -2e+34) {
		tmp = 3.0 * (sqrt(x) * (-1.0 + y));
	} else if (t_1 <= 1e+153) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * 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) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * 3.0d0
    t_1 = ((-1.0d0) + (y + (1.0d0 / (x * 9.0d0)))) * t_0
    if (t_1 <= (-2d+34)) then
        tmp = 3.0d0 * (sqrt(x) * ((-1.0d0) + y))
    else if (t_1 <= 1d+153) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = y * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * 3.0;
	double t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	double tmp;
	if (t_1 <= -2e+34) {
		tmp = 3.0 * (Math.sqrt(x) * (-1.0 + y));
	} else if (t_1 <= 1e+153) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * 3.0
	t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0
	tmp = 0
	if t_1 <= -2e+34:
		tmp = 3.0 * (math.sqrt(x) * (-1.0 + y))
	elif t_1 <= 1e+153:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = y * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(-1.0 + Float64(y + Float64(1.0 / Float64(x * 9.0)))) * t_0)
	tmp = 0.0
	if (t_1 <= -2e+34)
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(-1.0 + y)));
	elseif (t_1 <= 1e+153)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(y * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * 3.0;
	t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	tmp = 0.0;
	if (t_1 <= -2e+34)
		tmp = 3.0 * (sqrt(x) * (-1.0 + y));
	elseif (t_1 <= 1e+153)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = y * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 + N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+34], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+153], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -1.99999999999999989e34

   1. Initial program 99.6%

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

      1. 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 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 99.6

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

   7. Simplified 99.6%

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

   if -1.99999999999999989e34 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 1e153

   1. Initial program 99.2%

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

   3. Taylor expanded in 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 86.7

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

   5. Simplified 86.7%

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

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

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 95.4

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

   5. Simplified 95.4%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -2 \cdot 10^{+34}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(-1 + y\right)\right)\\ \mathbf{elif}\;\left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(-1 + y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (+ -1.0 (+ y (/ 1.0 (* x 9.0)))) t_0)))
   (if (<= t_1 -10.0)
     (* 3.0 (* (sqrt x) (+ -1.0 y)))
     (if (<= t_1 4e+153) (/ (* (sqrt x) 0.3333333333333333) x) (* y t_0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = 3.0 * (sqrt(x) * (-1.0 + y));
	} else if (t_1 <= 4e+153) {
		tmp = (sqrt(x) * 0.3333333333333333) / x;
	} else {
		tmp = y * 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) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * 3.0d0
    t_1 = ((-1.0d0) + (y + (1.0d0 / (x * 9.0d0)))) * t_0
    if (t_1 <= (-10.0d0)) then
        tmp = 3.0d0 * (sqrt(x) * ((-1.0d0) + y))
    else if (t_1 <= 4d+153) then
        tmp = (sqrt(x) * 0.3333333333333333d0) / x
    else
        tmp = y * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * 3.0;
	double t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = 3.0 * (Math.sqrt(x) * (-1.0 + y));
	} else if (t_1 <= 4e+153) {
		tmp = (Math.sqrt(x) * 0.3333333333333333) / x;
	} else {
		tmp = y * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * 3.0
	t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0
	tmp = 0
	if t_1 <= -10.0:
		tmp = 3.0 * (math.sqrt(x) * (-1.0 + y))
	elif t_1 <= 4e+153:
		tmp = (math.sqrt(x) * 0.3333333333333333) / x
	else:
		tmp = y * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(-1.0 + Float64(y + Float64(1.0 / Float64(x * 9.0)))) * t_0)
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(-1.0 + y)));
	elseif (t_1 <= 4e+153)
		tmp = Float64(Float64(sqrt(x) * 0.3333333333333333) / x);
	else
		tmp = Float64(y * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * 3.0;
	t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	tmp = 0.0;
	if (t_1 <= -10.0)
		tmp = 3.0 * (sqrt(x) * (-1.0 + y));
	elseif (t_1 <= 4e+153)
		tmp = (sqrt(x) * 0.3333333333333333) / x;
	else
		tmp = y * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 + N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+153], N[(N[(N[Sqrt[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision], N[(y * t$95$0), $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))) < -10

   1. Initial program 99.6%

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

      1. 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 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 98.8

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

   7. Simplified 98.8%

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

   if -10 < (*.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))) < 4e153

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

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

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

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

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

      17. sqrt-lowering-sqrt.f64 92.2

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

   5. Simplified 92.2%

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

   6. Taylor expanded in x around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 84.8

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

   8. Simplified 84.8%

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

   if 4e153 < (*.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 inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -10:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(-1 + y\right)\right)\\ \mathbf{elif}\;\left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(-1 + y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (+ -1.0 (+ y (/ 1.0 (* x 9.0)))) t_0)))
   (if (<= t_1 -10.0)
     (* 3.0 (* (sqrt x) (+ -1.0 y)))
     (if (<= t_1 4e+153) (/ 0.3333333333333333 (sqrt x)) (* y t_0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = 3.0 * (sqrt(x) * (-1.0 + y));
	} else if (t_1 <= 4e+153) {
		tmp = 0.3333333333333333 / sqrt(x);
	} else {
		tmp = y * 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) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * 3.0d0
    t_1 = ((-1.0d0) + (y + (1.0d0 / (x * 9.0d0)))) * t_0
    if (t_1 <= (-10.0d0)) then
        tmp = 3.0d0 * (sqrt(x) * ((-1.0d0) + y))
    else if (t_1 <= 4d+153) then
        tmp = 0.3333333333333333d0 / sqrt(x)
    else
        tmp = y * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * 3.0;
	double t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = 3.0 * (Math.sqrt(x) * (-1.0 + y));
	} else if (t_1 <= 4e+153) {
		tmp = 0.3333333333333333 / Math.sqrt(x);
	} else {
		tmp = y * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * 3.0
	t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0
	tmp = 0
	if t_1 <= -10.0:
		tmp = 3.0 * (math.sqrt(x) * (-1.0 + y))
	elif t_1 <= 4e+153:
		tmp = 0.3333333333333333 / math.sqrt(x)
	else:
		tmp = y * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(-1.0 + Float64(y + Float64(1.0 / Float64(x * 9.0)))) * t_0)
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(-1.0 + y)));
	elseif (t_1 <= 4e+153)
		tmp = Float64(0.3333333333333333 / sqrt(x));
	else
		tmp = Float64(y * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * 3.0;
	t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	tmp = 0.0;
	if (t_1 <= -10.0)
		tmp = 3.0 * (sqrt(x) * (-1.0 + y));
	elseif (t_1 <= 4e+153)
		tmp = 0.3333333333333333 / sqrt(x);
	else
		tmp = y * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 + N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+153], N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(y * t$95$0), $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))) < -10

   1. Initial program 99.6%

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

      1. 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 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 98.8

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

   7. Simplified 98.8%

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

   if -10 < (*.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))) < 4e153

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. /-lowering-/.f64 84.7

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

   5. Simplified 84.7%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

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

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

      5. sqrt-lowering-sqrt.f64 84.7

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

   7. Applied egg-rr 84.7%

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

   if 4e153 < (*.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 inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -10:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(-1 + y\right)\right)\\ \mathbf{elif}\;\left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (+ -1.0 (+ y (/ 1.0 (* x 9.0)))) t_0)))
   (if (<= t_1 -10.0)
     (* (sqrt x) (fma 3.0 y -3.0))
     (if (<= t_1 4e+153) (/ 0.3333333333333333 (sqrt x)) (* y t_0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (-1.0 + (y + (1.0 / (x * 9.0)))) * t_0;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = sqrt(x) * fma(3.0, y, -3.0);
	} else if (t_1 <= 4e+153) {
		tmp = 0.3333333333333333 / sqrt(x);
	} else {
		tmp = y * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(-1.0 + Float64(y + Float64(1.0 / Float64(x * 9.0)))) * t_0)
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = Float64(sqrt(x) * fma(3.0, y, -3.0));
	elseif (t_1 <= 4e+153)
		tmp = Float64(0.3333333333333333 / sqrt(x));
	else
		tmp = Float64(y * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 + N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+153], N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(y * t$95$0), $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))) < -10

   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 98.8

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

   5. Simplified 98.8%

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

   if -10 < (*.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))) < 4e153

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. /-lowering-/.f64 84.7

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

   5. Simplified 84.7%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

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

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

      5. sqrt-lowering-sqrt.f64 84.7

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

   7. Applied egg-rr 84.7%

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

   if 4e153 < (*.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 inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 99.3

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

   5. Simplified 99.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -10:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision] + y), $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. 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 99.4

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

4. Applied egg-rr 99.4%

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

5. Final simplification 99.4%

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

Alternative 8: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. distribute-rgt-in N/A

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

   15. metadata-eval N/A

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

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

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

   17. associate-*r/ N/A

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

   18. metadata-eval N/A

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

   19. associate-*l/ N/A

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

   20. metadata-eval N/A

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

   21. /-lowering-/.f64 99.1

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

5. Simplified 99.1%

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

Alternative 9: 60.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.75e-9)
   (* y (* (sqrt x) 3.0))
   (if (<= y 1.9e-13) (* (sqrt x) -3.0) (* (* (sqrt x) y) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.75e-9) {
		tmp = y * (sqrt(x) * 3.0);
	} else if (y <= 1.9e-13) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = (sqrt(x) * y) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.75d-9)) then
        tmp = y * (sqrt(x) * 3.0d0)
    else if (y <= 1.9d-13) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = (sqrt(x) * y) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.75e-9) {
		tmp = y * (Math.sqrt(x) * 3.0);
	} else if (y <= 1.9e-13) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = (Math.sqrt(x) * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.75e-9:
		tmp = y * (math.sqrt(x) * 3.0)
	elif y <= 1.9e-13:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = (math.sqrt(x) * y) * 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.75e-9)
		tmp = Float64(y * Float64(sqrt(x) * 3.0));
	elseif (y <= 1.9e-13)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(Float64(sqrt(x) * y) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.75e-9)
		tmp = y * (sqrt(x) * 3.0);
	elseif (y <= 1.9e-13)
		tmp = sqrt(x) * -3.0;
	else
		tmp = (sqrt(x) * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.75e-9], N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e-13], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.75e-9

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 68.2

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

   5. Simplified 68.2%

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

   if -1.75e-9 < y < 1.9e-13

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 50.0

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

   8. Simplified 50.0%

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

   if 1.9e-13 < y

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 70.9

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

   5. Simplified 70.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 71.0

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

   7. Applied egg-rr 71.0%

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

Alternative 10: 60.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;y \leq -1.75 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* (sqrt x) 3.0))))
   (if (<= y -1.75e-9) t_0 (if (<= y 1.9e-13) (* (sqrt x) -3.0) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (sqrt(x) * 3.0);
	double tmp;
	if (y <= -1.75e-9) {
		tmp = t_0;
	} else if (y <= 1.9e-13) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (sqrt(x) * 3.0d0)
    if (y <= (-1.75d-9)) then
        tmp = t_0
    else if (y <= 1.9d-13) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (Math.sqrt(x) * 3.0);
	double tmp;
	if (y <= -1.75e-9) {
		tmp = t_0;
	} else if (y <= 1.9e-13) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (math.sqrt(x) * 3.0)
	tmp = 0
	if y <= -1.75e-9:
		tmp = t_0
	elif y <= 1.9e-13:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(sqrt(x) * 3.0))
	tmp = 0.0
	if (y <= -1.75e-9)
		tmp = t_0;
	elseif (y <= 1.9e-13)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (sqrt(x) * 3.0);
	tmp = 0.0;
	if (y <= -1.75e-9)
		tmp = t_0;
	elseif (y <= 1.9e-13)
		tmp = sqrt(x) * -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.75e-9], t$95$0, If[LessEqual[y, 1.9e-13], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.75e-9 or 1.9e-13 < 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. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 69.3

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

   5. Simplified 69.3%

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

   if -1.75e-9 < y < 1.9e-13

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 50.0

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

   8. Simplified 50.0%

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

Alternative 11: 61.6% 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 60.0

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

5. Simplified 60.0%

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

Alternative 12: 25.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 64.3

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

5. Simplified 64.3%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. sqrt-lowering-sqrt.f64 26.0

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

8. Simplified 26.0%

   \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
9. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024205 
(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)))