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

Percentage Accurate: 99.4% → 99.5%

Time: 7.7s

Alternatives: 10

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. 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. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

   10. cancel-sign-sub-inv N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. distribute-rgt-out-- N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-sqrt.f64 99.5

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.6%

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

Alternative 2: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_1 \leq 10^{+153}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x)))
        (t_1 (* t_0 (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_1 -1.0)
     (* (fma 3.0 y -3.0) (sqrt x))
     (if (<= t_1 1e+153)
       (* (sqrt (pow x -1.0)) 0.3333333333333333)
       (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = fma(3.0, y, -3.0) * sqrt(x);
	} else if (t_1 <= 1e+153) {
		tmp = sqrt(pow(x, -1.0)) * 0.3333333333333333;
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(fma(3.0, y, -3.0) * sqrt(x));
	elseif (t_1 <= 1e+153)
		tmp = Float64(sqrt((x ^ -1.0)) * 0.3333333333333333);
	else
		tmp = Float64(t_0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(3.0 * y + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+153], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(t$95$0 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if -1 < (*.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))) < 1e153

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 1e153 < (*.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 99.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 10^{+153}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \sqrt{x}\\ t_1 := t\_0 \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_1 \leq -2000000:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_1 \leq 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (sqrt x)))
        (t_1 (* t_0 (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_1 -2000000.0)
     (* (fma 3.0 y -3.0) (sqrt x))
     (if (<= t_1 1e+153)
       (* (sqrt x) (- (/ 0.3333333333333333 x) 3.0))
       (* t_0 y)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * sqrt(x);
	double t_1 = t_0 * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_1 <= -2000000.0) {
		tmp = fma(3.0, y, -3.0) * sqrt(x);
	} else if (t_1 <= 1e+153) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) - 3.0);
	} else {
		tmp = t_0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * sqrt(x))
	t_1 = Float64(t_0 * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_1 <= -2000000.0)
		tmp = Float64(fma(3.0, y, -3.0) * sqrt(x));
	elseif (t_1 <= 1e+153)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) - 3.0));
	else
		tmp = Float64(t_0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2000000.0], N[(N[(3.0 * y + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+153], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -2e6

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if -2e6 < (*.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))) < 1e153

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 82.3%

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

   if 1e153 < (*.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 99.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -2000000:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 10^{+153}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-in N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

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

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

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

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

   8. mul-1-neg N/A

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

   9. rem-square-sqrt N/A

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

   10. unpow2 N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. rem-square-sqrt N/A

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

   16. *-commutative N/A

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

   17. distribute-lft1-in N/A

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

   18. +-commutative N/A

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

   19. lower-fma.f64 N/A

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

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

Alternative 5: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-y, -3, \frac{0.3333333333333333}{x}\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, y, -3\right) \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3e-8)
   (* (fma (- y) -3.0 (/ 0.3333333333333333 x)) (sqrt x))
   (* (fma 3.0 y -3.0) (sqrt x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3e-8) {
		tmp = fma(-y, -3.0, (0.3333333333333333 / x)) * sqrt(x);
	} else {
		tmp = fma(3.0, y, -3.0) * sqrt(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3e-8)
		tmp = Float64(fma(Float64(-y), -3.0, Float64(0.3333333333333333 / x)) * sqrt(x));
	else
		tmp = Float64(fma(3.0, y, -3.0) * sqrt(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3e-8], N[(N[((-y) * -3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * y + -3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.99999999999999973e-8

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. rem-square-sqrt N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. rem-square-sqrt N/A

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

      16. *-commutative N/A

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

      17. distribute-lft1-in N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 98.6%

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

   if 2.99999999999999973e-8 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 6: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-in N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

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

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

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

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

   8. mul-1-neg N/A

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

   9. rem-square-sqrt N/A

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

   10. unpow2 N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. rem-square-sqrt N/A

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

   16. *-commutative N/A

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

   17. distribute-lft1-in N/A

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

   18. +-commutative N/A

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

   19. lower-fma.f64 N/A

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

7. Applied rewrites 99.5%

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

Alternative 7: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

5. Applied rewrites 99.5%

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

Alternative 8: 62.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-sqrt.f64 65.7

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

5. Applied rewrites 65.7%

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

Alternative 9: 38.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 40.6

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

5. Applied rewrites 40.6%

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

7. Applied rewrites 40.6%

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

Alternative 10: 38.7% accurate, 2.0× speedup

Math

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

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(Float64(3.0 * sqrt(x)) * y)
end

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 40.6

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

5. Applied rewrites 40.6%

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

7. Applied rewrites 40.6%

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{y} \]
8. 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 2024313 
(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)))