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

Percentage Accurate: 99.4% → 99.4%

Time: 7.5s

Alternatives: 14

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return ((((-1.0 / x) / -9.0) + 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 = (((((-1.0d0) / x) / (-9.0d0)) + y) - 1.0d0) * (sqrt(x) * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. neg-mul-1 N/A

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

   6. lower-/.f64 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 92.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ t_1 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(\frac{1}{x \cdot 3} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma y 3.0 -3.0) (sqrt x)))
        (t_1 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) (* (sqrt x) 3.0))))
   (if (<= t_1 -2e+27)
     t_0
     (if (<= t_1 5e+152) (* (+ (/ 1.0 (* x 3.0)) -3.0) (sqrt x)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, 3.0, -3.0) * sqrt(x);
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * (sqrt(x) * 3.0);
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = t_0;
	} else if (t_1 <= 5e+152) {
		tmp = ((1.0 / (x * 3.0)) + -3.0) * sqrt(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(y, 3.0, -3.0) * sqrt(x))
	t_1 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * Float64(sqrt(x) * 3.0))
	tmp = 0.0
	if (t_1 <= -2e+27)
		tmp = t_0;
	elseif (t_1 <= 5e+152)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * 3.0)) + -3.0) * sqrt(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 3.0 + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+27], t$95$0, If[LessEqual[t$95$1, 5e+152], N[(N[(N[(1.0 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -2e27 or 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 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 99.6

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

   5. Applied rewrites 99.6%

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

   if -2e27 < (*.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 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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. *-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 81.3

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

   5. Applied rewrites 81.3%

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

   7. Applied rewrites 81.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(\frac{1}{x \cdot 3} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ t_1 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{x} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma y 3.0 -3.0) (sqrt x)))
        (t_1 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) (* (sqrt x) 3.0))))
   (if (<= t_1 -2e+27)
     t_0
     (if (<= t_1 5e+152) (* (+ (/ 0.3333333333333333 x) -3.0) (sqrt x)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, 3.0, -3.0) * sqrt(x);
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * (sqrt(x) * 3.0);
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = t_0;
	} else if (t_1 <= 5e+152) {
		tmp = ((0.3333333333333333 / x) + -3.0) * sqrt(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(y, 3.0, -3.0) * sqrt(x))
	t_1 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * Float64(sqrt(x) * 3.0))
	tmp = 0.0
	if (t_1 <= -2e+27)
		tmp = t_0;
	elseif (t_1 <= 5e+152)
		tmp = Float64(Float64(Float64(0.3333333333333333 / x) + -3.0) * sqrt(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 3.0 + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+27], t$95$0, If[LessEqual[t$95$1, 5e+152], N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -2e27 or 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 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 99.6

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

   5. Applied rewrites 99.6%

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

   if -2e27 < (*.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 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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. *-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 81.3

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

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333}{x} + -3\right) \cdot \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{x} + -3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ t_1 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma y 3.0 -3.0) (sqrt x)))
        (t_1 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) (* (sqrt x) 3.0))))
   (if (<= t_1 -200.0) t_0 (if (<= t_1 5e+152) (/ (sqrt x) (* x 3.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, 3.0, -3.0) * sqrt(x);
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * (sqrt(x) * 3.0);
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+152) {
		tmp = sqrt(x) / (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(y, 3.0, -3.0) * sqrt(x))
	t_1 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * Float64(sqrt(x) * 3.0))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_0;
	elseif (t_1 <= 5e+152)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 3.0 + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$0, If[LessEqual[t$95$1, 5e+152], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -200 or 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 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 98.4

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

   5. Applied rewrites 98.4%

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

   if -200 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.3

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt{\frac{1}{x}}} \]
   6. 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 79.7

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

   7. Applied rewrites 79.7%

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

   9. Applied rewrites 79.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -200:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ t_1 := \left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma y 3.0 -3.0) (sqrt x)))
        (t_1 (* (- (+ (/ 1.0 (* 9.0 x)) y) 1.0) (* (sqrt x) 3.0))))
   (if (<= t_1 -200.0)
     t_0
     (if (<= t_1 5e+152) (/ 0.3333333333333333 (sqrt x)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, 3.0, -3.0) * sqrt(x);
	double t_1 = (((1.0 / (9.0 * x)) + y) - 1.0) * (sqrt(x) * 3.0);
	double tmp;
	if (t_1 <= -200.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+152) {
		tmp = 0.3333333333333333 / sqrt(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(y, 3.0, -3.0) * sqrt(x))
	t_1 = Float64(Float64(Float64(Float64(1.0 / Float64(9.0 * x)) + y) - 1.0) * Float64(sqrt(x) * 3.0))
	tmp = 0.0
	if (t_1 <= -200.0)
		tmp = t_0;
	elseif (t_1 <= 5e+152)
		tmp = Float64(0.3333333333333333 / sqrt(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 3.0 + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200.0], t$95$0, If[LessEqual[t$95$1, 5e+152], N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -200 or 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 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 98.4

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

   5. Applied rewrites 98.4%

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

   if -200 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.3

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt{\frac{1}{x}}} \]
   6. 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 79.7

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

   7. Applied rewrites 79.7%

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

   9. Applied rewrites 79.7%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq -200:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(\left(\frac{1}{9 \cdot x} + y\right) - 1\right) \cdot \left(\sqrt{x} \cdot 3\right) \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3, -3\right) \cdot \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (((1.0 / (9.0 * x)) + 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 = (((1.0d0 / (9.0d0 * x)) + y) - 1.0d0) * (sqrt(x) * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 7: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. frac-2neg N/A

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

   9. div-inv N/A

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

   10. neg-mul-1 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. un-div-inv N/A

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

   17. lower-/.f64 N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. lower--.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 8: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return ((((0.1111111111111111 / x) + 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 = ((((0.1111111111111111d0 / x) + y) - 1.0d0) * sqrt(x)) * 3.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.5

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 9: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. 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. Final simplification 99.4%

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

Alternative 10: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -95:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;y \leq 0.57:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \sqrt{x}\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -95.0)
   (* y (* (sqrt x) 3.0))
   (if (<= y 0.57) (* -3.0 (sqrt x)) (* (* y (sqrt x)) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -95.0) {
		tmp = y * (sqrt(x) * 3.0);
	} else if (y <= 0.57) {
		tmp = -3.0 * sqrt(x);
	} else {
		tmp = (y * sqrt(x)) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-95.0d0)) then
        tmp = y * (sqrt(x) * 3.0d0)
    else if (y <= 0.57d0) then
        tmp = (-3.0d0) * sqrt(x)
    else
        tmp = (y * sqrt(x)) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -95.0) {
		tmp = y * (Math.sqrt(x) * 3.0);
	} else if (y <= 0.57) {
		tmp = -3.0 * Math.sqrt(x);
	} else {
		tmp = (y * Math.sqrt(x)) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -95.0:
		tmp = y * (math.sqrt(x) * 3.0)
	elif y <= 0.57:
		tmp = -3.0 * math.sqrt(x)
	else:
		tmp = (y * math.sqrt(x)) * 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -95.0)
		tmp = Float64(y * Float64(sqrt(x) * 3.0));
	elseif (y <= 0.57)
		tmp = Float64(-3.0 * sqrt(x));
	else
		tmp = Float64(Float64(y * sqrt(x)) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -95.0)
		tmp = y * (sqrt(x) * 3.0);
	elseif (y <= 0.57)
		tmp = -3.0 * sqrt(x);
	else
		tmp = (y * sqrt(x)) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -95.0], N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.57], N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -95

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 78.0%

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

   if -95 < y < 0.569999999999999951

   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. *-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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.3%

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

   if 0.569999999999999951 < 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 82.5

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

   5. Applied rewrites 82.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -95:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;y \leq 0.57:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \sqrt{x}\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -95:\\ \;\;\;\;\left(y \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 0.57:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \sqrt{x}\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -95.0)
   (* (* y 3.0) (sqrt x))
   (if (<= y 0.57) (* -3.0 (sqrt x)) (* (* y (sqrt x)) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -95.0) {
		tmp = (y * 3.0) * sqrt(x);
	} else if (y <= 0.57) {
		tmp = -3.0 * sqrt(x);
	} else {
		tmp = (y * sqrt(x)) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-95.0d0)) then
        tmp = (y * 3.0d0) * sqrt(x)
    else if (y <= 0.57d0) then
        tmp = (-3.0d0) * sqrt(x)
    else
        tmp = (y * sqrt(x)) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -95.0) {
		tmp = (y * 3.0) * Math.sqrt(x);
	} else if (y <= 0.57) {
		tmp = -3.0 * Math.sqrt(x);
	} else {
		tmp = (y * Math.sqrt(x)) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -95.0:
		tmp = (y * 3.0) * math.sqrt(x)
	elif y <= 0.57:
		tmp = -3.0 * math.sqrt(x)
	else:
		tmp = (y * math.sqrt(x)) * 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -95.0)
		tmp = Float64(Float64(y * 3.0) * sqrt(x));
	elseif (y <= 0.57)
		tmp = Float64(-3.0 * sqrt(x));
	else
		tmp = Float64(Float64(y * sqrt(x)) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -95.0)
		tmp = (y * 3.0) * sqrt(x);
	elseif (y <= 0.57)
		tmp = -3.0 * sqrt(x);
	else
		tmp = (y * sqrt(x)) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -95.0], N[(N[(y * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.57], N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -95

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 78.0%

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

   if -95 < y < 0.569999999999999951

   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. *-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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.3%

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

   if 0.569999999999999951 < 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 82.5

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

   5. Applied rewrites 82.5%

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

Alternative 12: 61.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -95:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.57:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 3.0) (sqrt x))))
   (if (<= y -95.0) t_0 (if (<= y 0.57) (* -3.0 (sqrt x)) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 3.0) * sqrt(x);
	double tmp;
	if (y <= -95.0) {
		tmp = t_0;
	} else if (y <= 0.57) {
		tmp = -3.0 * sqrt(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * 3.0d0) * sqrt(x)
    if (y <= (-95.0d0)) then
        tmp = t_0
    else if (y <= 0.57d0) then
        tmp = (-3.0d0) * sqrt(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 3.0) * Math.sqrt(x);
	double tmp;
	if (y <= -95.0) {
		tmp = t_0;
	} else if (y <= 0.57) {
		tmp = -3.0 * Math.sqrt(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * 3.0) * math.sqrt(x)
	tmp = 0
	if y <= -95.0:
		tmp = t_0
	elif y <= 0.57:
		tmp = -3.0 * math.sqrt(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 3.0) * sqrt(x))
	tmp = 0.0
	if (y <= -95.0)
		tmp = t_0;
	elseif (y <= 0.57)
		tmp = Float64(-3.0 * sqrt(x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * 3.0) * sqrt(x);
	tmp = 0.0;
	if (y <= -95.0)
		tmp = t_0;
	elseif (y <= 0.57)
		tmp = -3.0 * sqrt(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -95.0], t$95$0, If[LessEqual[y, 0.57], N[(-3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -95 or 0.569999999999999951 < 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 80.4

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 80.3%

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

   if -95 < y < 0.569999999999999951

   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. *-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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.3%

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

Alternative 13: 62.2% 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 99.5%

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

3. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-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 67.3

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

5. Applied rewrites 67.3%

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

Alternative 14: 25.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. 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 62.0

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

5. Applied rewrites 62.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 30.0%

   \[\leadsto -3 \cdot \sqrt{\color{blue}{x}} \]
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 2024271 
(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)))