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

Percentage Accurate: 99.4% → 99.4%

Time: 8.8s

Alternatives: 13

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.4

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. associate-/r* N/A

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

   14. *-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/r* N/A

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

   18. metadata-eval N/A

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

   19. lift-/.f64 N/A

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

   20. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 89.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\left({\left(x \cdot 3\right)}^{-1} - 3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x)))
        (t_1 (* t_0 (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_1 -5e+77)
     (* (fma 3.0 y -3.0) (sqrt x))
     (if (<= t_1 2e+149)
       (* (- (pow (* x 3.0) -1.0) 3.0) (sqrt x))
       (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -5e+77) {
		tmp = fma(3.0, y, -3.0) * sqrt(x);
	} else if (t_1 <= 2e+149) {
		tmp = (pow((x * 3.0), -1.0) - 3.0) * sqrt(x);
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_1 <= -5e+77)
		tmp = Float64(fma(3.0, y, -3.0) * sqrt(x));
	elseif (t_1 <= 2e+149)
		tmp = Float64(Float64((Float64(x * 3.0) ^ -1.0) - 3.0) * sqrt(x));
	else
		tmp = Float64(t_0 * y);
	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[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+77], N[(N[(3.0 * y + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+149], N[(N[(N[Power[N[(x * 3.0), $MachinePrecision], -1.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -5.00000000000000004e77

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -5.00000000000000004e77 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 2.0000000000000001e149

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 90.6

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

   7. Applied rewrites 90.6%

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

   9. Applied rewrites 90.6%

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

   if 2.0000000000000001e149 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\left({\left(x \cdot 3\right)}^{-1} - 3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x)))
        (t_1 (* t_0 (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_1 -10.0)
     (* (fma 3.0 y -3.0) (sqrt x))
     (if (<= t_1 2e+149)
       (* (sqrt (pow x -1.0)) 0.3333333333333333)
       (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -10.0) {
		tmp = fma(3.0, y, -3.0) * sqrt(x);
	} else if (t_1 <= 2e+149) {
		tmp = sqrt(pow(x, -1.0)) * 0.3333333333333333;
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = Float64(fma(3.0, y, -3.0) * sqrt(x));
	elseif (t_1 <= 2e+149)
		tmp = Float64(sqrt((x ^ -1.0)) * 0.3333333333333333);
	else
		tmp = Float64(t_0 * y);
	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[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], N[(N[(3.0 * y + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+149], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(t$95$0 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 98.0

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

   5. Applied rewrites 98.0%

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

   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))) < 2.0000000000000001e149

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if 2.0000000000000001e149 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -10:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x)))
        (t_1 (* t_0 (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_1 -5e+77)
     (* (fma 3.0 y -3.0) (sqrt x))
     (if (<= t_1 2e+149)
       (* (sqrt x) (- (/ 0.3333333333333333 x) 3.0))
       (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -5e+77) {
		tmp = fma(3.0, y, -3.0) * sqrt(x);
	} else if (t_1 <= 2e+149) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) - 3.0);
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_1 <= -5e+77)
		tmp = Float64(fma(3.0, y, -3.0) * sqrt(x));
	elseif (t_1 <= 2e+149)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) - 3.0));
	else
		tmp = Float64(t_0 * y);
	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[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+77], N[(N[(3.0 * y + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+149], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -5.00000000000000004e77

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -5.00000000000000004e77 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 2.0000000000000001e149

   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. distribute-lft-out-- N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-out-- N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 90.6%

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

   if 2.0000000000000001e149 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.008:\\ \;\;\;\;\mathsf{fma}\left(-y, -3, \frac{0.3333333333333333}{x}\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.008)
   (* (fma (- y) -3.0 (/ 0.3333333333333333 x)) (sqrt x))
   (* (* (- y 1.0) (sqrt x)) 3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.008) {
		tmp = fma(-y, -3.0, (0.3333333333333333 / x)) * sqrt(x);
	} else {
		tmp = ((y - 1.0) * sqrt(x)) * 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.008)
		tmp = Float64(fma(Float64(-y), -3.0, Float64(0.3333333333333333 / x)) * sqrt(x));
	else
		tmp = Float64(Float64(Float64(y - 1.0) * sqrt(x)) * 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.008], N[(N[((-y) * -3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0080000000000000002

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-rgt-neg-out N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-in N/A

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

      17. rem-square-sqrt N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

      20. unpow2 N/A

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

      21. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 98.6%

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

   if 0.0080000000000000002 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 98.9

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

   7. Applied rewrites 98.9%

      \[\leadsto \left(\color{blue}{\left(y - 1\right)} \cdot \sqrt{x}\right) \cdot 3 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.4

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

Alternative 7: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-in N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

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

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

   8. distribute-rgt-neg-out N/A

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

   9. mul-1-neg N/A

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

   10. metadata-eval N/A

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

   11. distribute-lft-in N/A

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

   12. distribute-rgt-in N/A

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

   13. metadata-eval N/A

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

   14. distribute-lft1-in N/A

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

   15. metadata-eval N/A

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

   16. distribute-lft-in N/A

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

   17. rem-square-sqrt N/A

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

   18. metadata-eval N/A

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

   19. sub-neg N/A

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

   20. unpow2 N/A

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

   21. lower-fma.f64 N/A

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

7. Applied rewrites 99.5%

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

Alternative 8: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

5. Applied rewrites 99.5%

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

Alternative 9: 61.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8000000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8000000000000.0) (not (<= y 1.0)))
   (* (* (sqrt x) y) 3.0)
   (* -3.0 (sqrt x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8000000000000.0) || !(y <= 1.0)) {
		tmp = (sqrt(x) * y) * 3.0;
	} else {
		tmp = -3.0 * sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8000000000000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = (sqrt(x) * y) * 3.0d0
    else
        tmp = (-3.0d0) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8000000000000.0) || !(y <= 1.0)) {
		tmp = (Math.sqrt(x) * y) * 3.0;
	} else {
		tmp = -3.0 * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8000000000000.0) or not (y <= 1.0):
		tmp = (math.sqrt(x) * y) * 3.0
	else:
		tmp = -3.0 * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8000000000000.0) || !(y <= 1.0))
		tmp = Float64(Float64(sqrt(x) * y) * 3.0);
	else
		tmp = Float64(-3.0 * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8000000000000.0) || ~((y <= 1.0)))
		tmp = (sqrt(x) * y) * 3.0;
	else
		tmp = -3.0 * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8000000000000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $MachinePrecision], N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8e12 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -8e12 < y < 1

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 45.9%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8000000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8000000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8000000000000.0) (not (<= y 1.0)))
   (* (* 3.0 y) (sqrt x))
   (* -3.0 (sqrt x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8000000000000.0) || !(y <= 1.0)) {
		tmp = (3.0 * y) * sqrt(x);
	} else {
		tmp = -3.0 * sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8000000000000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = (3.0d0 * y) * sqrt(x)
    else
        tmp = (-3.0d0) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8000000000000.0) || !(y <= 1.0)) {
		tmp = (3.0 * y) * Math.sqrt(x);
	} else {
		tmp = -3.0 * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8000000000000.0) or not (y <= 1.0):
		tmp = (3.0 * y) * math.sqrt(x)
	else:
		tmp = -3.0 * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8000000000000.0) || !(y <= 1.0))
		tmp = Float64(Float64(3.0 * y) * sqrt(x));
	else
		tmp = Float64(-3.0 * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8000000000000.0) || ~((y <= 1.0)))
		tmp = (3.0 * y) * sqrt(x);
	else
		tmp = -3.0 * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8000000000000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8e12 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 75.9

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 75.9%

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

   if -8e12 < y < 1

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 45.9%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8000000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8000000000000:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8000000000000.0)
   (* (* 3.0 y) (sqrt x))
   (if (<= y 1.0) (* -3.0 (sqrt x)) (* (* 3.0 (sqrt x)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8000000000000.0) {
		tmp = (3.0 * y) * sqrt(x);
	} else if (y <= 1.0) {
		tmp = -3.0 * sqrt(x);
	} else {
		tmp = (3.0 * sqrt(x)) * 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 <= (-8000000000000.0d0)) then
        tmp = (3.0d0 * y) * sqrt(x)
    else if (y <= 1.0d0) then
        tmp = (-3.0d0) * sqrt(x)
    else
        tmp = (3.0d0 * sqrt(x)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8000000000000.0) {
		tmp = (3.0 * y) * Math.sqrt(x);
	} else if (y <= 1.0) {
		tmp = -3.0 * Math.sqrt(x);
	} else {
		tmp = (3.0 * Math.sqrt(x)) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8000000000000.0:
		tmp = (3.0 * y) * math.sqrt(x)
	elif y <= 1.0:
		tmp = -3.0 * math.sqrt(x)
	else:
		tmp = (3.0 * math.sqrt(x)) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8000000000000.0)
		tmp = Float64(Float64(3.0 * y) * sqrt(x));
	elseif (y <= 1.0)
		tmp = Float64(-3.0 * sqrt(x));
	else
		tmp = Float64(Float64(3.0 * sqrt(x)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8000000000000.0)
		tmp = (3.0 * y) * sqrt(x);
	elseif (y <= 1.0)
		tmp = -3.0 * sqrt(x);
	else
		tmp = (3.0 * sqrt(x)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8000000000000.0], N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8e12

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 74.2

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 74.2%

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

   if -8e12 < y < 1

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 45.9%

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

   if 1 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 77.6%

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

Alternative 12: 63.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-sqrt.f64 60.5

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

5. Applied rewrites 60.5%

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

Alternative 13: 26.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-in N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 64.3

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

7. Applied rewrites 64.3%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 26.2%

    \[\leadsto -3 \cdot \sqrt{x} \]
11. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024324 
(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)))