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

Percentage Accurate: 99.4% → 99.4%

Time: 10.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 2: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t\_0 \cdot \left(y + -1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x))) (t_1 (* t_0 (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_1 -5e+19)
     (* t_0 (+ y -1.0))
     (if (<= t_1 5e+151)
       (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
       (* (sqrt x) (* 3.0 y))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -5e+19) {
		tmp = t_0 * (y + -1.0);
	} else if (t_1 <= 5e+151) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 * sqrt(x)
    t_1 = t_0 * ((y + (1.0d0 / (x * 9.0d0))) + (-1.0d0))
    if (t_1 <= (-5d+19)) then
        tmp = t_0 * (y + (-1.0d0))
    else if (t_1 <= 5d+151) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * Math.sqrt(x);
	double t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -5e+19) {
		tmp = t_0 * (y + -1.0);
	} else if (t_1 <= 5e+151) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * math.sqrt(x)
	t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0)
	tmp = 0
	if t_1 <= -5e+19:
		tmp = t_0 * (y + -1.0)
	elif t_1 <= 5e+151:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_1 <= -5e+19)
		tmp = Float64(t_0 * Float64(y + -1.0));
	elseif (t_1 <= 5e+151)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * sqrt(x);
	t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	tmp = 0.0;
	if (t_1 <= -5e+19)
		tmp = t_0 * (y + -1.0);
	elseif (t_1 <= 5e+151)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+19], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+151], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -5e19

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 98.3%

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

   if -5e19 < (*.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))) < 5.0000000000000002e151

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 87.1

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

   5. Simplified 87.1%

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

   if 5.0000000000000002e151 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 99.4

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

   5. Simplified 99.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 99.6

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_0 \cdot \left(y + -1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x))) (t_1 (* t_0 (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_1 -0.1)
     (* t_0 (+ y -1.0))
     (if (<= t_1 5e+151)
       (* 0.3333333333333333 (sqrt (/ 1.0 x)))
       (* (sqrt x) (* 3.0 y))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -0.1) {
		tmp = t_0 * (y + -1.0);
	} else if (t_1 <= 5e+151) {
		tmp = 0.3333333333333333 * sqrt((1.0 / x));
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 * sqrt(x)
    t_1 = t_0 * ((y + (1.0d0 / (x * 9.0d0))) + (-1.0d0))
    if (t_1 <= (-0.1d0)) then
        tmp = t_0 * (y + (-1.0d0))
    else if (t_1 <= 5d+151) then
        tmp = 0.3333333333333333d0 * sqrt((1.0d0 / x))
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * Math.sqrt(x);
	double t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -0.1) {
		tmp = t_0 * (y + -1.0);
	} else if (t_1 <= 5e+151) {
		tmp = 0.3333333333333333 * Math.sqrt((1.0 / x));
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * math.sqrt(x)
	t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0)
	tmp = 0
	if t_1 <= -0.1:
		tmp = t_0 * (y + -1.0)
	elif t_1 <= 5e+151:
		tmp = 0.3333333333333333 * math.sqrt((1.0 / x))
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(t_0 * Float64(y + -1.0));
	elseif (t_1 <= 5e+151)
		tmp = Float64(0.3333333333333333 * sqrt(Float64(1.0 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * sqrt(x);
	t_1 = t_0 * ((y + (1.0 / (x * 9.0))) + -1.0);
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = t_0 * (y + -1.0);
	elseif (t_1 <= 5e+151)
		tmp = 0.3333333333333333 * sqrt((1.0 / x));
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+151], N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -0.10000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 95.9%

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

   if -0.10000000000000001 < (*.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))) < 5.0000000000000002e151

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. /-lowering-/.f64 85.7

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

   5. Simplified 85.7%

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

   if 5.0000000000000002e151 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 99.4

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

   5. Simplified 99.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 99.6

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -0.1:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 5 \cdot 10^{+151}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (* (sqrt x) (fma 3.0 y (/ 0.3333333333333333 x)))
   (* (* 3.0 (sqrt x)) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = sqrt(x) * fma(3.0, y, (0.3333333333333333 / x));
	} else {
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(sqrt(x) * fma(3.0, y, Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(Float64(3.0 * sqrt(x)) * Float64(y + -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.4%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval N/A

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

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

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in y around 0

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. metadata-eval N/A

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

   7. Simplified 99.3%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 98.2

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

   10. Simplified 98.2%

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

       1. *-commutative N/A

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

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

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

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

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

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

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

       5. metadata-eval N/A

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

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

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

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

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

       8. sqrt-lowering-sqrt.f64 98.2

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

   12. Applied egg-rr 98.2%

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

   if 0.110000000000000001 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 97.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

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

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

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

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

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

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

   16. metadata-eval N/A

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

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

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

   18. *-commutative N/A

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

   19. associate-*l* N/A

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

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

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in y around 0

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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

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

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

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

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

   9. distribute-lft-out N/A

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

   10. +-commutative N/A

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

   11. distribute-lft-out N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

   15. distribute-lft-in N/A

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

   16. metadata-eval N/A

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

7. Simplified 99.4%

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

Alternative 6: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{-9}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 0.122:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.55e-9)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 0.122) (* (sqrt x) -3.0) (* (sqrt x) (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.55e-9) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 0.122) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.55d-9)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 0.122d0) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.55e-9) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 0.122) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.55e-9:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 0.122:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.55e-9)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 0.122)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.55e-9)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 0.122)
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.55e-9], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.122], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.54999999999999994e-9

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 75.0

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

   5. Simplified 75.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 75.0

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

   7. Applied egg-rr 75.0%

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

   if -3.54999999999999994e-9 < y < 0.122

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 48.3

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

   8. Simplified 48.3%

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

   if 0.122 < y

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 74.2

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

   5. Simplified 74.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 74.3

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

   7. Applied egg-rr 74.3%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{-9}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 0.122:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{-9}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \mathbf{elif}\;y \leq 0.122:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.55e-9)
   (* (* 3.0 (sqrt x)) y)
   (if (<= y 0.122) (* (sqrt x) -3.0) (* (sqrt x) (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.55e-9) {
		tmp = (3.0 * sqrt(x)) * y;
	} else if (y <= 0.122) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.55d-9)) then
        tmp = (3.0d0 * sqrt(x)) * y
    else if (y <= 0.122d0) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.55e-9) {
		tmp = (3.0 * Math.sqrt(x)) * y;
	} else if (y <= 0.122) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.55e-9:
		tmp = (3.0 * math.sqrt(x)) * y
	elif y <= 0.122:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.55e-9)
		tmp = Float64(Float64(3.0 * sqrt(x)) * y);
	elseif (y <= 0.122)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.55e-9)
		tmp = (3.0 * sqrt(x)) * y;
	elseif (y <= 0.122)
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.55e-9], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 0.122], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.54999999999999994e-9

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 75.0

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

   5. Simplified 75.0%

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

   if -3.54999999999999994e-9 < y < 0.122

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 48.3

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

   8. Simplified 48.3%

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

   if 0.122 < y

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 74.2

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

   5. Simplified 74.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 74.3

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

   7. Applied egg-rr 74.3%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{-9}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \mathbf{elif}\;y \leq 0.122:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot y\\ \mathbf{if}\;y \leq -3.55 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.122:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) y)))
   (if (<= y -3.55e-9) t_0 (if (<= y 0.122) (* (sqrt x) -3.0) t_0))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * y;
	double tmp;
	if (y <= -3.55e-9) {
		tmp = t_0;
	} else if (y <= 0.122) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (3.0d0 * sqrt(x)) * y
    if (y <= (-3.55d-9)) then
        tmp = t_0
    else if (y <= 0.122d0) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (3.0 * Math.sqrt(x)) * y;
	double tmp;
	if (y <= -3.55e-9) {
		tmp = t_0;
	} else if (y <= 0.122) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (3.0 * math.sqrt(x)) * y
	tmp = 0
	if y <= -3.55e-9:
		tmp = t_0
	elif y <= 0.122:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * y)
	tmp = 0.0
	if (y <= -3.55e-9)
		tmp = t_0;
	elseif (y <= 0.122)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (3.0 * sqrt(x)) * y;
	tmp = 0.0;
	if (y <= -3.55e-9)
		tmp = t_0;
	elseif (y <= 0.122)
		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 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -3.55e-9], t$95$0, If[LessEqual[y, 0.122], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.54999999999999994e-9 or 0.122 < y

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. sqrt-lowering-sqrt.f64 74.6

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

   5. Simplified 74.6%

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

   if -3.54999999999999994e-9 < y < 0.122

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 48.3

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

   8. Simplified 48.3%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{-9}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \mathbf{elif}\;y \leq 0.122:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in y around inf

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

5. Simplified 62.1%

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

6. Final simplification 62.1%

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

Alternative 10: 62.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-in N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 62.0

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

5. Simplified 62.0%

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

Alternative 11: 25.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. metadata-eval N/A

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

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

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. associate-*l/ N/A

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

   15. metadata-eval N/A

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

   16. /-lowering-/.f64 61.6

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

5. Simplified 61.6%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. sqrt-lowering-sqrt.f64 25.4

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

8. Simplified 25.4%

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

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(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)))