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

Percentage Accurate: 99.4% → 99.4%

Time: 9.0s

Alternatives: 11

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. 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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in y around 0

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. distribute-lft-in N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. associate-*r* N/A

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

   18. *-commutative N/A

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

   19. associate-*r* N/A

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\left(y - 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{x} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) t_0)))
   (if (<= t_1 -2e+37)
     (* (- y 1.0) t_0)
     (if (<= t_1 5e+152)
       (* (+ (/ 0.3333333333333333 x) -3.0) (sqrt x))
       (* (* (sqrt x) y) 3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -2e+37) {
		tmp = (y - 1.0) * t_0;
	} else if (t_1 <= 5e+152) {
		tmp = ((0.3333333333333333 / x) + -3.0) * sqrt(x);
	} else {
		tmp = (sqrt(x) * y) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * 3.0d0
    t_1 = (((1.0d0 / (9.0d0 * x)) + y) - 1.0d0) * t_0
    if (t_1 <= (-2d+37)) then
        tmp = (y - 1.0d0) * t_0
    else if (t_1 <= 5d+152) then
        tmp = ((0.3333333333333333d0 / x) + (-3.0d0)) * sqrt(x)
    else
        tmp = (sqrt(x) * y) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -2e+37) {
		tmp = (y - 1.0) * t_0;
	} else if (t_1 <= 5e+152) {
		tmp = ((0.3333333333333333 / x) + -3.0) * Math.sqrt(x);
	} else {
		tmp = (Math.sqrt(x) * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * 3.0
	t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0
	tmp = 0
	if t_1 <= -2e+37:
		tmp = (y - 1.0) * t_0
	elif t_1 <= 5e+152:
		tmp = ((0.3333333333333333 / x) + -3.0) * math.sqrt(x)
	else:
		tmp = (math.sqrt(x) * y) * 3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * t_0)
	tmp = 0.0
	if (t_1 <= -2e+37)
		tmp = Float64(Float64(y - 1.0) * t_0);
	elseif (t_1 <= 5e+152)
		tmp = Float64(Float64(Float64(0.3333333333333333 / x) + -3.0) * sqrt(x));
	else
		tmp = Float64(Float64(sqrt(x) * y) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * 3.0;
	t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	tmp = 0.0;
	if (t_1 <= -2e+37)
		tmp = (y - 1.0) * t_0;
	elseif (t_1 <= 5e+152)
		tmp = ((0.3333333333333333 / x) + -3.0) * sqrt(x);
	else
		tmp = (sqrt(x) * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+37], N[(N[(y - 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e+152], N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -1.99999999999999991e37

   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

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

      1. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -1.99999999999999991e37 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 5e152

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if 5e152 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{x} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\left(y - 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) t_0)))
   (if (<= t_1 -0.05)
     (* (- y 1.0) t_0)
     (if (<= t_1 5e+152)
       (* 0.3333333333333333 (sqrt (/ 1.0 x)))
       (* (* (sqrt x) y) 3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (y - 1.0) * t_0;
	} else if (t_1 <= 5e+152) {
		tmp = 0.3333333333333333 * sqrt((1.0 / x));
	} else {
		tmp = (sqrt(x) * y) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * 3.0d0
    t_1 = (((1.0d0 / (9.0d0 * x)) + y) - 1.0d0) * t_0
    if (t_1 <= (-0.05d0)) then
        tmp = (y - 1.0d0) * t_0
    else if (t_1 <= 5d+152) then
        tmp = 0.3333333333333333d0 * sqrt((1.0d0 / x))
    else
        tmp = (sqrt(x) * y) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (y - 1.0) * t_0;
	} else if (t_1 <= 5e+152) {
		tmp = 0.3333333333333333 * Math.sqrt((1.0 / x));
	} else {
		tmp = (Math.sqrt(x) * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * 3.0
	t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0
	tmp = 0
	if t_1 <= -0.05:
		tmp = (y - 1.0) * t_0
	elif t_1 <= 5e+152:
		tmp = 0.3333333333333333 * math.sqrt((1.0 / x))
	else:
		tmp = (math.sqrt(x) * y) * 3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * t_0)
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(Float64(y - 1.0) * t_0);
	elseif (t_1 <= 5e+152)
		tmp = Float64(0.3333333333333333 * sqrt(Float64(1.0 / x)));
	else
		tmp = Float64(Float64(sqrt(x) * y) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * 3.0;
	t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	tmp = 0.0;
	if (t_1 <= -0.05)
		tmp = (y - 1.0) * t_0;
	elseif (t_1 <= 5e+152)
		tmp = 0.3333333333333333 * sqrt((1.0 / x));
	else
		tmp = (sqrt(x) * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.05], N[(N[(y - 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e+152], N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -0.050000000000000003

   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

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

      1. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 5e152

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

      1. *-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 84.1

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

   5. Applied rewrites 84.1%

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

   if 5e152 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -0.05:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ t_1 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\left(y - 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{0.3333333333333333}{x} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)) (t_1 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) t_0)))
   (if (<= t_1 -0.05)
     (* (- y 1.0) t_0)
     (if (<= t_1 5e+152)
       (* (/ 0.3333333333333333 x) (sqrt x))
       (* (* (sqrt x) y) 3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (y - 1.0) * t_0;
	} else if (t_1 <= 5e+152) {
		tmp = (0.3333333333333333 / x) * sqrt(x);
	} else {
		tmp = (sqrt(x) * y) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * 3.0d0
    t_1 = (((1.0d0 / (9.0d0 * x)) + y) - 1.0d0) * t_0
    if (t_1 <= (-0.05d0)) then
        tmp = (y - 1.0d0) * t_0
    else if (t_1 <= 5d+152) then
        tmp = (0.3333333333333333d0 / x) * sqrt(x)
    else
        tmp = (sqrt(x) * y) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * 3.0;
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (y - 1.0) * t_0;
	} else if (t_1 <= 5e+152) {
		tmp = (0.3333333333333333 / x) * Math.sqrt(x);
	} else {
		tmp = (Math.sqrt(x) * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * 3.0
	t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0
	tmp = 0
	if t_1 <= -0.05:
		tmp = (y - 1.0) * t_0
	elif t_1 <= 5e+152:
		tmp = (0.3333333333333333 / x) * math.sqrt(x)
	else:
		tmp = (math.sqrt(x) * y) * 3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	t_1 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * t_0)
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(Float64(y - 1.0) * t_0);
	elseif (t_1 <= 5e+152)
		tmp = Float64(Float64(0.3333333333333333 / x) * sqrt(x));
	else
		tmp = Float64(Float64(sqrt(x) * y) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * 3.0;
	t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * t_0;
	tmp = 0.0;
	if (t_1 <= -0.05)
		tmp = (y - 1.0) * t_0;
	elseif (t_1 <= 5e+152)
		tmp = (0.3333333333333333 / x) * sqrt(x);
	else
		tmp = (sqrt(x) * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.05], N[(N[(y - 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e+152], N[(N[(0.3333333333333333 / x), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -0.050000000000000003

   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

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

      1. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 5e152

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. +-commutative N/A

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

      13. associate--l+ N/A

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

      14. +-commutative N/A

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

      15. associate--l+ N/A

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

      16. +-commutative N/A

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

      17. distribute-rgt-in N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.1%

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

   if 5e152 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -0.05:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{0.3333333333333333}{x} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. 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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in y around 0

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. distribute-lft-in N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. associate-*r* N/A

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

   18. *-commutative N/A

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

   19. associate-*r* N/A

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

11. Applied rewrites 99.5%

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

Alternative 6: 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 99.4%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-in N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. associate--l+ N/A

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

   16. +-commutative N/A

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

   17. distribute-rgt-in N/A

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 7: 61.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-8)
   (* (* (sqrt x) y) 3.0)
   (if (<= y 7.5e-18) (* (sqrt x) -3.0) (* (* 3.0 y) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-8) {
		tmp = (sqrt(x) * y) * 3.0;
	} else if (y <= 7.5e-18) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = (3.0 * y) * 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 <= (-2.2d-8)) then
        tmp = (sqrt(x) * y) * 3.0d0
    else if (y <= 7.5d-18) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = (3.0d0 * y) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-8) {
		tmp = (Math.sqrt(x) * y) * 3.0;
	} else if (y <= 7.5e-18) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = (3.0 * y) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e-8:
		tmp = (math.sqrt(x) * y) * 3.0
	elif y <= 7.5e-18:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = (3.0 * y) * math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e-8)
		tmp = Float64(Float64(sqrt(x) * y) * 3.0);
	elseif (y <= 7.5e-18)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(Float64(3.0 * y) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e-8)
		tmp = (sqrt(x) * y) * 3.0;
	elseif (y <= 7.5e-18)
		tmp = sqrt(x) * -3.0;
	else
		tmp = (3.0 * y) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e-8], N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[y, 7.5e-18], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1999999999999998e-8

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if -2.1999999999999998e-8 < y < 7.50000000000000015e-18

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   7. Applied rewrites 47.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 47.0%

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

   if 7.50000000000000015e-18 < 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 63.6

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

   5. Applied rewrites 63.6%

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

   7. Applied rewrites 63.7%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;\left(\sqrt{x} \cdot y\right) \cdot 3\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot y\right) \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 y) (sqrt x))))
   (if (<= y -2.2e-8) t_0 (if (<= y 7.5e-18) (* (sqrt x) -3.0) t_0))))

C

double code(double x, double y) {
	double t_0 = (3.0 * y) * sqrt(x);
	double tmp;
	if (y <= -2.2e-8) {
		tmp = t_0;
	} else if (y <= 7.5e-18) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (3.0d0 * y) * sqrt(x)
    if (y <= (-2.2d-8)) then
        tmp = t_0
    else if (y <= 7.5d-18) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (3.0 * y) * Math.sqrt(x);
	double tmp;
	if (y <= -2.2e-8) {
		tmp = t_0;
	} else if (y <= 7.5e-18) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (3.0 * y) * math.sqrt(x)
	tmp = 0
	if y <= -2.2e-8:
		tmp = t_0
	elif y <= 7.5e-18:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * y) * sqrt(x))
	tmp = 0.0
	if (y <= -2.2e-8)
		tmp = t_0;
	elseif (y <= 7.5e-18)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (3.0 * y) * sqrt(x);
	tmp = 0.0;
	if (y <= -2.2e-8)
		tmp = t_0;
	elseif (y <= 7.5e-18)
		tmp = sqrt(x) * -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * y), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e-8], t$95$0, If[LessEqual[y, 7.5e-18], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1999999999999998e-8 or 7.50000000000000015e-18 < 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 66.0%

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

   if -2.1999999999999998e-8 < y < 7.50000000000000015e-18

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

   7. Applied rewrites 47.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 47.0%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (y - 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 - 1.0d0) * (sqrt(x) * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around inf

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

   1. lower--.f64 57.1

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

5. Applied rewrites 57.1%

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

6. Final simplification 57.1%

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

Alternative 10: 62.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around inf

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

   1. 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-rgt-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-sqrt.f64 57.1

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

5. Applied rewrites 57.1%

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

Alternative 11: 25.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft-in N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. associate-*r* N/A

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

   14. associate-*r* N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

7. Applied rewrites 53.7%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 24.5%

    \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]
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 2024270 
(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)))