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

Percentage Accurate: 99.4% → 99.4%

Time: 9.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sqrt.f64 69.7

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

5. Applied rewrites 69.7%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_0 \leq -400000000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(-3, \sqrt{x}, \frac{0.3333333333333333}{\sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_0 -400000000.0)
     (* (* (- y 1.0) 3.0) (sqrt x))
     (if (<= t_0 5e+143)
       (fma -3.0 (sqrt x) (/ 0.3333333333333333 (sqrt x)))
       (* (- y 1.0) (* (sqrt x) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -400000000.0) {
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	} else if (t_0 <= 5e+143) {
		tmp = fma(-3.0, sqrt(x), (0.3333333333333333 / sqrt(x)));
	} else {
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_0 <= -400000000.0)
		tmp = Float64(Float64(Float64(y - 1.0) * 3.0) * sqrt(x));
	elseif (t_0 <= 5e+143)
		tmp = fma(-3.0, sqrt(x), Float64(0.3333333333333333 / sqrt(x)));
	else
		tmp = Float64(Float64(y - 1.0) * Float64(sqrt(x) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400000000.0], N[(N[(N[(y - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+143], N[(-3.0 * N[Sqrt[x], $MachinePrecision] + N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -4e8

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

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -4e8 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 5.00000000000000012e143

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.4%

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

   10. Applied rewrites 99.4%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 88.6%

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

   if 5.00000000000000012e143 < (*.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. 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. lower-*.f64 99.5

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 93.8

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

   7. Applied rewrites 93.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -400000000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(-3, \sqrt{x}, \frac{0.3333333333333333}{\sqrt{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\left(\left(\frac{0.1111111111111111}{x} - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_0 -5e+27)
     (* (* (- y 1.0) 3.0) (sqrt x))
     (if (<= t_0 5e+143)
       (* (* (- (/ 0.1111111111111111 x) 1.0) (sqrt x)) 3.0)
       (* (- y 1.0) (* (sqrt x) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -5e+27) {
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	} else if (t_0 <= 5e+143) {
		tmp = (((0.1111111111111111 / x) - 1.0) * sqrt(x)) * 3.0;
	} else {
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (3.0d0 * sqrt(x)) * ((y + ((x * 9.0d0) ** (-1.0d0))) - 1.0d0)
    if (t_0 <= (-5d+27)) then
        tmp = ((y - 1.0d0) * 3.0d0) * sqrt(x)
    else if (t_0 <= 5d+143) then
        tmp = (((0.1111111111111111d0 / x) - 1.0d0) * sqrt(x)) * 3.0d0
    else
        tmp = (y - 1.0d0) * (sqrt(x) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (3.0 * Math.sqrt(x)) * ((y + Math.pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -5e+27) {
		tmp = ((y - 1.0) * 3.0) * Math.sqrt(x);
	} else if (t_0 <= 5e+143) {
		tmp = (((0.1111111111111111 / x) - 1.0) * Math.sqrt(x)) * 3.0;
	} else {
		tmp = (y - 1.0) * (Math.sqrt(x) * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (3.0 * math.sqrt(x)) * ((y + math.pow((x * 9.0), -1.0)) - 1.0)
	tmp = 0
	if t_0 <= -5e+27:
		tmp = ((y - 1.0) * 3.0) * math.sqrt(x)
	elif t_0 <= 5e+143:
		tmp = (((0.1111111111111111 / x) - 1.0) * math.sqrt(x)) * 3.0
	else:
		tmp = (y - 1.0) * (math.sqrt(x) * 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_0 <= -5e+27)
		tmp = Float64(Float64(Float64(y - 1.0) * 3.0) * sqrt(x));
	elseif (t_0 <= 5e+143)
		tmp = Float64(Float64(Float64(Float64(0.1111111111111111 / x) - 1.0) * sqrt(x)) * 3.0);
	else
		tmp = Float64(Float64(y - 1.0) * Float64(sqrt(x) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (3.0 * sqrt(x)) * ((y + ((x * 9.0) ^ -1.0)) - 1.0);
	tmp = 0.0;
	if (t_0 <= -5e+27)
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	elseif (t_0 <= 5e+143)
		tmp = (((0.1111111111111111 / x) - 1.0) * sqrt(x)) * 3.0;
	else
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+27], N[(N[(N[(y - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+143], N[(N[(N[(N[(0.1111111111111111 / x), $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision], N[(N[(y - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -4.99999999999999979e27

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

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -4.99999999999999979e27 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 5.00000000000000012e143

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 89.5

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

   7. Applied rewrites 89.5%

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

   if 5.00000000000000012e143 < (*.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. 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. lower-*.f64 99.5

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 93.8

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

   7. Applied rewrites 93.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\left(\left(\frac{0.1111111111111111}{x} - 1\right) \cdot \sqrt{x}\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\left(\left(\frac{0.1111111111111111}{x} - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_0 -5e+27)
     (* (* (- y 1.0) 3.0) (sqrt x))
     (if (<= t_0 5e+143)
       (* (* (- (/ 0.1111111111111111 x) 1.0) 3.0) (sqrt x))
       (* (- y 1.0) (* (sqrt x) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -5e+27) {
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	} else if (t_0 <= 5e+143) {
		tmp = (((0.1111111111111111 / x) - 1.0) * 3.0) * sqrt(x);
	} else {
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (3.0d0 * sqrt(x)) * ((y + ((x * 9.0d0) ** (-1.0d0))) - 1.0d0)
    if (t_0 <= (-5d+27)) then
        tmp = ((y - 1.0d0) * 3.0d0) * sqrt(x)
    else if (t_0 <= 5d+143) then
        tmp = (((0.1111111111111111d0 / x) - 1.0d0) * 3.0d0) * sqrt(x)
    else
        tmp = (y - 1.0d0) * (sqrt(x) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (3.0 * Math.sqrt(x)) * ((y + Math.pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -5e+27) {
		tmp = ((y - 1.0) * 3.0) * Math.sqrt(x);
	} else if (t_0 <= 5e+143) {
		tmp = (((0.1111111111111111 / x) - 1.0) * 3.0) * Math.sqrt(x);
	} else {
		tmp = (y - 1.0) * (Math.sqrt(x) * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (3.0 * math.sqrt(x)) * ((y + math.pow((x * 9.0), -1.0)) - 1.0)
	tmp = 0
	if t_0 <= -5e+27:
		tmp = ((y - 1.0) * 3.0) * math.sqrt(x)
	elif t_0 <= 5e+143:
		tmp = (((0.1111111111111111 / x) - 1.0) * 3.0) * math.sqrt(x)
	else:
		tmp = (y - 1.0) * (math.sqrt(x) * 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_0 <= -5e+27)
		tmp = Float64(Float64(Float64(y - 1.0) * 3.0) * sqrt(x));
	elseif (t_0 <= 5e+143)
		tmp = Float64(Float64(Float64(Float64(0.1111111111111111 / x) - 1.0) * 3.0) * sqrt(x));
	else
		tmp = Float64(Float64(y - 1.0) * Float64(sqrt(x) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (3.0 * sqrt(x)) * ((y + ((x * 9.0) ^ -1.0)) - 1.0);
	tmp = 0.0;
	if (t_0 <= -5e+27)
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	elseif (t_0 <= 5e+143)
		tmp = (((0.1111111111111111 / x) - 1.0) * 3.0) * sqrt(x);
	else
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+27], N[(N[(N[(y - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+143], N[(N[(N[(N[(0.1111111111111111 / x), $MachinePrecision] - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(y - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -4.99999999999999979e27

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

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -4.99999999999999979e27 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 5.00000000000000012e143

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if 5.00000000000000012e143 < (*.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. 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. lower-*.f64 99.5

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 93.8

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

   7. Applied rewrites 93.8%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\left(\left(\frac{0.1111111111111111}{x} - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_0 \leq -100000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_0 -100000.0)
     (* (* (- y 1.0) 3.0) (sqrt x))
     (if (<= t_0 2e+152)
       (sqrt (/ 0.1111111111111111 x))
       (* (- y 1.0) (* (sqrt x) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -100000.0) {
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	} else if (t_0 <= 2e+152) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (3.0d0 * sqrt(x)) * ((y + ((x * 9.0d0) ** (-1.0d0))) - 1.0d0)
    if (t_0 <= (-100000.0d0)) then
        tmp = ((y - 1.0d0) * 3.0d0) * sqrt(x)
    else if (t_0 <= 2d+152) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = (y - 1.0d0) * (sqrt(x) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (3.0 * Math.sqrt(x)) * ((y + Math.pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -100000.0) {
		tmp = ((y - 1.0) * 3.0) * Math.sqrt(x);
	} else if (t_0 <= 2e+152) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = (y - 1.0) * (Math.sqrt(x) * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (3.0 * math.sqrt(x)) * ((y + math.pow((x * 9.0), -1.0)) - 1.0)
	tmp = 0
	if t_0 <= -100000.0:
		tmp = ((y - 1.0) * 3.0) * math.sqrt(x)
	elif t_0 <= 2e+152:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = (y - 1.0) * (math.sqrt(x) * 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_0 <= -100000.0)
		tmp = Float64(Float64(Float64(y - 1.0) * 3.0) * sqrt(x));
	elseif (t_0 <= 2e+152)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(Float64(y - 1.0) * Float64(sqrt(x) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (3.0 * sqrt(x)) * ((y + ((x * 9.0) ^ -1.0)) - 1.0);
	tmp = 0.0;
	if (t_0 <= -100000.0)
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	elseif (t_0 <= 2e+152)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = (y - 1.0) * (sqrt(x) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100000.0], N[(N[(N[(y - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+152], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], N[(N[(y - 1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -1e5

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

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -1e5 < (*.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))) < 2.0000000000000001e152

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 86.4

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

   7. Applied rewrites 86.4%

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

   9. Applied rewrites 86.7%

      \[\leadsto \sqrt{\frac{0.1111111111111111}{x}} \]

   if 2.0000000000000001e152 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. 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. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower--.f64 99.6

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

   7. Applied rewrites 99.6%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -100000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - 1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_0 \leq -100000 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+143}\right):\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(y, 3, -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (or (<= t_0 -100000.0) (not (<= t_0 5e+143)))
     (* (sqrt x) (fma y 3.0 -3.0))
     (sqrt (/ 0.1111111111111111 x)))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if ((t_0 <= -100000.0) || !(t_0 <= 5e+143)) {
		tmp = sqrt(x) * fma(y, 3.0, -3.0);
	} else {
		tmp = sqrt((0.1111111111111111 / x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if ((t_0 <= -100000.0) || !(t_0 <= 5e+143))
		tmp = Float64(sqrt(x) * fma(y, 3.0, -3.0));
	else
		tmp = sqrt(Float64(0.1111111111111111 / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -100000.0], N[Not[LessEqual[t$95$0, 5e+143]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0 + -3.0), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]]]

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))) < -1e5 or 5.00000000000000012e143 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64)))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x around inf

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

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

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

       2. *-rgt-identity N/A

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

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

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

       4. metadata-eval N/A

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

       5. fp-cancel-sign-sub-inv N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-lft-out N/A

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

       11. lower-*.f64 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 96.7

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

   11. Applied rewrites 96.7%

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

   if -1e5 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 5.00000000000000012e143

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 87.6

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

   7. Applied rewrites 87.6%

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

   9. Applied rewrites 87.9%

      \[\leadsto \sqrt{\frac{0.1111111111111111}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -100000 \lor \neg \left(\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 5 \cdot 10^{+143}\right):\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(y, 3, -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right)\\ \mathbf{if}\;t\_0 \leq -100000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(y, 3, -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 3.0 (sqrt x)) (- (+ y (pow (* x 9.0) -1.0)) 1.0))))
   (if (<= t_0 -100000.0)
     (* (* (- y 1.0) 3.0) (sqrt x))
     (if (<= t_0 5e+143)
       (sqrt (/ 0.1111111111111111 x))
       (* (sqrt x) (fma y 3.0 -3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 * sqrt(x)) * ((y + pow((x * 9.0), -1.0)) - 1.0);
	double tmp;
	if (t_0 <= -100000.0) {
		tmp = ((y - 1.0) * 3.0) * sqrt(x);
	} else if (t_0 <= 5e+143) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = sqrt(x) * fma(y, 3.0, -3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + (Float64(x * 9.0) ^ -1.0)) - 1.0))
	tmp = 0.0
	if (t_0 <= -100000.0)
		tmp = Float64(Float64(Float64(y - 1.0) * 3.0) * sqrt(x));
	elseif (t_0 <= 5e+143)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(sqrt(x) * fma(y, 3.0, -3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100000.0], N[(N[(N[(y - 1.0), $MachinePrecision] * 3.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+143], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0 + -3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < -1e5

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

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -1e5 < (*.f64 (*.f64 #s(literal 3 binary64) (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) #s(literal 1 binary64))) < 5.00000000000000012e143

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 87.6

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

   7. Applied rewrites 87.6%

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

   9. Applied rewrites 87.9%

      \[\leadsto \sqrt{\frac{0.1111111111111111}{x}} \]

   if 5.00000000000000012e143 < (*.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 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x around inf

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

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

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

       2. *-rgt-identity N/A

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

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

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

       4. metadata-eval N/A

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

       5. fp-cancel-sign-sub-inv N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. distribute-lft-out N/A

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

       11. lower-*.f64 N/A

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

       12. lower-sqrt.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 93.8

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

   11. Applied rewrites 93.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq -100000:\\ \;\;\;\;\left(\left(y - 1\right) \cdot 3\right) \cdot \sqrt{x}\\ \mathbf{elif}\;\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + {\left(x \cdot 9\right)}^{-1}\right) - 1\right) \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \mathsf{fma}\left(y, 3, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sqrt.f64 69.7

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

5. Applied rewrites 69.7%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.5%

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

Alternative 9: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (((y - ((-0.1111111111111111d0) / x)) - 1.0d0) * sqrt(x)) * 3.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.4%

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

Alternative 10: 62.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0 + -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 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sqrt.f64 69.7

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

5. Applied rewrites 69.7%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around inf

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

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

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

    2. *-rgt-identity N/A

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

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

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

    4. metadata-eval N/A

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

    5. fp-cancel-sign-sub-inv N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

    8. associate-*l* N/A

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

    9. *-commutative N/A

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

    10. distribute-lft-out N/A

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

    11. lower-*.f64 N/A

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

    12. lower-sqrt.f64 N/A

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

    13. *-commutative N/A

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

    14. lower-fma.f64 57.4

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

11. Applied rewrites 57.4%

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

Alternative 11: 38.8% 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 99.4%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 35.4

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

5. Applied rewrites 35.4%

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

7. Applied rewrites 35.4%

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

Alternative 12: 38.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in y around 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 35.4

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

5. Applied rewrites 35.4%

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

7. Applied rewrites 35.4%

   \[\leadsto \left(3 \cdot y\right) \cdot \color{blue}{\sqrt{x}} \]
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 2024329 
(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)))