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

Percentage Accurate: 99.4% → 99.4%

Time: 10.8s

Alternatives: 9

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 99.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 99.4

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

4. Applied rewrites 99.4%

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

Alternative 2: 91.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ t_1 := \sqrt{x} \cdot 3\\ t_2 := t\_1 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 10^{+152}:\\ \;\;\;\;t\_1 \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (fma 3.0 y -3.0)))
        (t_1 (* (sqrt x) 3.0))
        (t_2 (* t_1 (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_2 -4e+42)
     t_0
     (if (<= t_2 1e+152) (* t_1 (+ (/ 0.1111111111111111 x) -1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * fma(3.0, y, -3.0);
	double t_1 = sqrt(x) * 3.0;
	double t_2 = t_1 * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_2 <= -4e+42) {
		tmp = t_0;
	} else if (t_2 <= 1e+152) {
		tmp = t_1 * ((0.1111111111111111 / x) + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * fma(3.0, y, -3.0))
	t_1 = Float64(sqrt(x) * 3.0)
	t_2 = Float64(t_1 * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_2 <= -4e+42)
		tmp = t_0;
	elseif (t_2 <= 1e+152)
		tmp = Float64(t_1 * Float64(Float64(0.1111111111111111 / x) + -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+42], t$95$0, If[LessEqual[t$95$2, 1e+152], N[(t$95$1 * N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $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))) < -4.00000000000000018e42 or 1e152 < (*.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 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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. *-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. lower-*.f64 N/A

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

      4. lower-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. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -4.00000000000000018e42 < (*.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))) < 1e152

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 89.4

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

   5. Applied rewrites 89.4%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(-1 + \frac{0.1111111111111111}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -4 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 10^{+152}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ t_1 := \left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+152}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (fma 3.0 y -3.0)))
        (t_1 (* (* (sqrt x) 3.0) (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_1 -4e+42)
     t_0
     (if (<= t_1 1e+152) (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x))) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * fma(3.0, y, -3.0);
	double t_1 = (sqrt(x) * 3.0) * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -4e+42) {
		tmp = t_0;
	} else if (t_1 <= 1e+152) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * fma(3.0, y, -3.0))
	t_1 = Float64(Float64(sqrt(x) * 3.0) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_1 <= -4e+42)
		tmp = t_0;
	elseif (t_1 <= 1e+152)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+42], t$95$0, If[LessEqual[t$95$1, 1e+152], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $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))) < -4.00000000000000018e42 or 1e152 < (*.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 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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. *-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. lower-*.f64 N/A

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

      4. lower-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. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -4.00000000000000018e42 < (*.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))) < 1e152

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

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

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

      5. lower-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. lower-+.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. lower-/.f64 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq -4 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \mathbf{elif}\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \leq 10^{+152}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (fma 3.0 y -3.0)))
        (t_1 (* (* (sqrt x) 3.0) (+ (+ y (/ 1.0 (* x 9.0))) -1.0))))
   (if (<= t_1 -2.0)
     t_0
     (if (<= t_1 1e+152) (* 0.3333333333333333 (sqrt (/ 1.0 x))) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * fma(3.0, y, -3.0);
	double t_1 = (sqrt(x) * 3.0) * ((y + (1.0 / (x * 9.0))) + -1.0);
	double tmp;
	if (t_1 <= -2.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+152) {
		tmp = 0.3333333333333333 * sqrt((1.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * fma(3.0, y, -3.0))
	t_1 = Float64(Float64(sqrt(x) * 3.0) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
	tmp = 0.0
	if (t_1 <= -2.0)
		tmp = t_0;
	elseif (t_1 <= 1e+152)
		tmp = Float64(0.3333333333333333 * sqrt(Float64(1.0 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y + -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2.0], t$95$0, If[LessEqual[t$95$1, 1e+152], N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $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))) < -2 or 1e152 < (*.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 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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. *-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. lower-*.f64 N/A

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

      4. lower-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. lower-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -2 < (*.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))) < 1e152

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 87.6

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

   5. Applied rewrites 87.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

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

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 40.0

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

5. Applied rewrites 40.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 40.0

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

7. Applied rewrites 40.0%

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

8. Taylor expanded in y around 0

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

   1. distribute-lft-out N/A

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

   2. associate--l+ N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. lower-fma.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 99.4

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

10. Applied rewrites 99.4%

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

12. Applied rewrites 99.4%

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

13. Final simplification 99.4%

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

Alternative 6: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 40.0

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

5. Applied rewrites 40.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 40.0

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

7. Applied rewrites 40.0%

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

8. Taylor expanded in y around 0

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

   1. distribute-lft-out N/A

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

   2. associate--l+ N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. lower-fma.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. associate-*r/ N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 99.4

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 7: 62.3% 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. lower-*.f64 N/A

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

   4. lower-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. lower-fma.f64 60.5

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

5. Applied rewrites 60.5%

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

Alternative 8: 38.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 38.2

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

7. Applied rewrites 38.2%

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

Alternative 9: 38.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 38.2

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

5. Applied rewrites 38.2%

   \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot y\right)} \]
6. 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 2024221 
(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)))