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

Percentage Accurate: 99.4% → 99.4%

Time: 10.4s

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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 99.4%

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

   1. fma-def 99.4%

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

   2. *-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. sub-neg 99.4%

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

   5. associate-*r/ 99.5%

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

   6. metadata-eval 99.5%

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

   7. metadata-eval 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 99.4%

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

   1. associate-*r* 99.4%

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

   2. sub-neg 99.4%

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

   3. associate-*r/ 99.5%

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

   4. metadata-eval 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. distribute-rgt-in 99.5%

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

   8. distribute-lft-in 99.5%

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

   9. +-commutative 99.5%

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

   10. distribute-lft-in 99.5%

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

   11. metadata-eval 99.5%

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

   12. associate-*l* 99.5%

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

   13. fma-udef 99.5%

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

   14. metadata-eval 99.5%

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

   15. associate-*r/ 99.4%

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

   16. fma-def 99.4%

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

   17. associate-*l* 99.4%

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

   18. distribute-lft-in 99.4%

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

   19. associate-*l* 99.4%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{2 \cdot \left(y + -1\right) + 0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.6e-6)
   (sqrt (+ (* 2.0 (+ y -1.0)) (* 0.1111111111111111 (/ 1.0 x))))
   (* (sqrt x) (- (* 3.0 y) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.6e-6) {
		tmp = sqrt(((2.0 * (y + -1.0)) + (0.1111111111111111 * (1.0 / x))));
	} else {
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6.6d-6) then
        tmp = sqrt(((2.0d0 * (y + (-1.0d0))) + (0.1111111111111111d0 * (1.0d0 / x))))
    else
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.6e-6) {
		tmp = Math.sqrt(((2.0 * (y + -1.0)) + (0.1111111111111111 * (1.0 / x))));
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.6e-6:
		tmp = math.sqrt(((2.0 * (y + -1.0)) + (0.1111111111111111 * (1.0 / x))))
	else:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.6e-6)
		tmp = sqrt(Float64(Float64(2.0 * Float64(y + -1.0)) + Float64(0.1111111111111111 * Float64(1.0 / x))));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) - 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.6e-6)
		tmp = sqrt(((2.0 * (y + -1.0)) + (0.1111111111111111 * (1.0 / x))));
	else
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.6e-6], N[Sqrt[N[(N[(2.0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.1111111111111111 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.60000000000000034e-6

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. associate--l+ 99.2%

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

      3. distribute-rgt-in 99.2%

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

      4. remove-double-neg 99.2%

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

      5. distribute-lft-neg-in 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. mul-1-neg 99.2%

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

      8. metadata-eval 99.2%

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

      9. *-commutative 99.2%

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

      10. associate-/r* 99.3%

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

      11. distribute-neg-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/r/ 99.3%

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

      14. associate-/l/ 99.3%

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

      15. associate-/r/ 99.3%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right)} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 88.8%

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

      2. pow2 88.8%

         \[\leadsto \color{blue}{{\left(\sqrt{\left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right)}\right)}^{2}} \]

   5. Applied egg-rr 88.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(\sqrt{x} \cdot 3\right) \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right)}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 88.8%

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

      2. sqrt-unprod 84.5%

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

      3. *-commutative 84.5%

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

      4. *-commutative 84.5%

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

      5. swap-sqr 43.3%

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

      6. pow2 43.3%

         \[\leadsto \sqrt{\color{blue}{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2}} \cdot \left(\left(\sqrt{x} \cdot 3\right) \cdot \left(\sqrt{x} \cdot 3\right)\right)} \]

      7. swap-sqr 43.4%

         \[\leadsto \sqrt{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2} \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(3 \cdot 3\right)\right)}} \]

      8. add-sqr-sqrt 43.5%

         \[\leadsto \sqrt{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2} \cdot \left(\color{blue}{x} \cdot \left(3 \cdot 3\right)\right)} \]

      9. metadata-eval 43.5%

         \[\leadsto \sqrt{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2} \cdot \left(x \cdot \color{blue}{9}\right)} \]

   7. Applied egg-rr 43.5%

      \[\leadsto \color{blue}{\sqrt{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2} \cdot \left(x \cdot 9\right)}} \]

   8. Taylor expanded in x around 0 79.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(y - 1\right) + 0.1111111111111111 \cdot \frac{1}{x}}} \]

   if 6.60000000000000034e-6 < x

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{2 \cdot \left(y + -1\right) + 0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]

Alternative 4: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+262} \lor \neg \left(x \leq 2.8 \cdot 10^{+284}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0005)
   (* (sqrt x) (/ 0.3333333333333333 x))
   (if (or (<= x 2.1e+262) (not (<= x 2.8e+284)))
     (* (sqrt x) -3.0)
     (* (sqrt x) (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0005) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else if ((x <= 2.1e+262) || !(x <= 2.8e+284)) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0005d0) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else if ((x <= 2.1d+262) .or. (.not. (x <= 2.8d+284))) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0005) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else if ((x <= 2.1e+262) || !(x <= 2.8e+284)) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0005:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	elif (x <= 2.1e+262) or not (x <= 2.8e+284):
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0005)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	elseif ((x <= 2.1e+262) || !(x <= 2.8e+284))
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0005)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	elseif ((x <= 2.1e+262) || ~((x <= 2.8e+284)))
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0005], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 2.1e+262], N[Not[LessEqual[x, 2.8e+284]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 99.2%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 77.6%

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

   if 5.0000000000000001e-4 < x < 2.09999999999999989e262 or 2.79999999999999996e284 < 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. Step-by-step derivation

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. distribute-rgt-in 99.7%

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

      4. remove-double-neg 99.7%

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

      5. distribute-lft-neg-in 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. mul-1-neg 99.7%

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

      8. metadata-eval 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-/r* 99.7%

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

      11. distribute-neg-frac 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/r/ 99.7%

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

      14. associate-/l/ 99.7%

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

      15. associate-/r/ 99.7%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.1%

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

   5. Taylor expanded in y around 0 56.5%

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

      1. *-commutative 56.5%

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

   7. Simplified 56.5%

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

   if 2.09999999999999989e262 < x < 2.79999999999999996e284

   1. Initial program 98.9%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around inf 67.8%

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

      1. associate-*r* 68.0%

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

      2. *-commutative 68.0%

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

   6. Simplified 68.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+262} \lor \neg \left(x \leq 2.8 \cdot 10^{+284}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]

Alternative 5: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -265000000:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -265000000.0)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 3.1e+34)
     (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
     (* y (sqrt (* x 9.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -265000000.0) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 3.1e+34) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = y * sqrt((x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-265000000.0d0)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 3.1d+34) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = y * sqrt((x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -265000000.0) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 3.1e+34) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = y * Math.sqrt((x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -265000000.0:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 3.1e+34:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = y * math.sqrt((x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -265000000.0)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 3.1e+34)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -265000000.0)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 3.1e+34)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = y * sqrt((x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -265000000.0], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+34], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.65e8

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 72.8%

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

   if -2.65e8 < y < 3.09999999999999977e34

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 96.6%

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

      1. *-commutative 96.6%

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

      2. sub-neg 96.6%

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

      3. associate-*r/ 96.7%

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

      4. metadata-eval 96.7%

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

      5. metadata-eval 96.7%

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

   6. Simplified 96.7%

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

   if 3.09999999999999977e34 < y

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 77.0%

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

      1. add-sqr-sqrt 76.8%

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

      2. sqrt-unprod 40.6%

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

      3. *-commutative 40.6%

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

      4. *-commutative 40.6%

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

      5. swap-sqr 40.5%

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

      6. metadata-eval 40.5%

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

      7. swap-sqr 36.9%

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

      8. add-sqr-sqrt 36.8%

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

   6. Applied egg-rr 36.8%

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

      1. unpow2 36.8%

         \[\leadsto \sqrt{9 \cdot \left(x \cdot \color{blue}{{y}^{2}}\right)} \]

      2. associate-*r* 36.9%

         \[\leadsto \sqrt{\color{blue}{\left(9 \cdot x\right) \cdot {y}^{2}}} \]

      3. *-commutative 36.9%

         \[\leadsto \sqrt{\color{blue}{\left(x \cdot 9\right)} \cdot {y}^{2}} \]

      4. unpow2 36.9%

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

   8. Simplified 36.9%

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

      1. sqrt-unprod 48.6%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \sqrt{y \cdot y}} \]

      2. add-sqr-sqrt 48.4%

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

      3. sqrt-prod 76.4%

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

      4. add-sqr-sqrt 76.7%

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

      5. associate-*l* 76.7%

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

      6. pow1/2 76.7%

         \[\leadsto \sqrt{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \cdot \left(\sqrt{\sqrt{x \cdot 9}} \cdot y\right) \]

      7. sqrt-pow1 76.8%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{x \cdot 9}} \cdot y\right) \]

      8. metadata-eval 76.8%

         \[\leadsto {\left(x \cdot 9\right)}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{x \cdot 9}} \cdot y\right) \]

      9. pow1/2 76.8%

         \[\leadsto {\left(x \cdot 9\right)}^{0.25} \cdot \left(\sqrt{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \cdot y\right) \]

      10. sqrt-pow1 76.7%

          \[\leadsto {\left(x \cdot 9\right)}^{0.25} \cdot \left(\color{blue}{{\left(x \cdot 9\right)}^{\left(\frac{0.5}{2}\right)}} \cdot y\right) \]

      11. metadata-eval 76.7%

          \[\leadsto {\left(x \cdot 9\right)}^{0.25} \cdot \left({\left(x \cdot 9\right)}^{\color{blue}{0.25}} \cdot y\right) \]

   10. Applied egg-rr 76.7%

       \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.25} \cdot \left({\left(x \cdot 9\right)}^{0.25} \cdot y\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 76.7%

          \[\leadsto \color{blue}{\left({\left(x \cdot 9\right)}^{0.25} \cdot {\left(x \cdot 9\right)}^{0.25}\right) \cdot y} \]

       2. pow-sqr 77.1%

          \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{\left(2 \cdot 0.25\right)}} \cdot y \]

       3. metadata-eval 77.1%

          \[\leadsto {\left(x \cdot 9\right)}^{\color{blue}{0.5}} \cdot y \]

       4. unpow1/2 77.1%

          \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot y \]

   12. Simplified 77.1%

       \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -265000000:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \end{array} \]

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * ((3.0d0 * y) + ((0.3333333333333333d0 / x) + (-3.0d0)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x) * ((3.0 * y) + ((0.3333333333333333 / x) + -3.0));
}

Python

def code(x, y):
	return math.sqrt(x) * ((3.0 * y) + ((0.3333333333333333 / x) + -3.0))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * ((3.0 * y) + ((0.3333333333333333 / x) + -3.0));
end

Wolfram

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

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. distribute-lft-in 99.4%

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

   4. +-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*r* 99.4%

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

   7. cancel-sign-sub 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-*r* 99.4%

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

   10. *-commutative 99.4%

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

   11. distribute-rgt-out-- 99.4%

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

   12. distribute-lft-neg-in 99.4%

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

   13. cancel-sign-sub 99.4%

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

   14. +-commutative 99.4%

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

   15. *-commutative 99.4%

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

   16. distribute-rgt-in 99.4%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 7: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \sqrt{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (+ y (+ (/ 0.1111111111111111 x) -1.0)) (sqrt (* x 9.0))))

C

double code(double x, double y) {
	return (y + ((0.1111111111111111 / x) + -1.0)) * sqrt((x * 9.0));
}

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 * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return (y + ((0.1111111111111111 / x) + -1.0)) * Math.sqrt((x * 9.0));
}

Python

def code(x, y):
	return (y + ((0.1111111111111111 / x) + -1.0)) * math.sqrt((x * 9.0))

Julia

function code(x, y)
	return Float64(Float64(y + Float64(Float64(0.1111111111111111 / x) + -1.0)) * sqrt(Float64(x * 9.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (y + ((0.1111111111111111 / x) + -1.0)) * sqrt((x * 9.0));
end

Wolfram

code[x_, y_] := N[(N[(y + N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(x * 9.0), $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. Step-by-step derivation

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. distribute-rgt-in 99.4%

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

   4. remove-double-neg 99.4%

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

   5. distribute-lft-neg-in 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. mul-1-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. *-commutative 99.4%

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

   10. associate-/r* 99.4%

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

   11. distribute-neg-frac 99.4%

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

   12. *-commutative 99.4%

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

   13. associate-/r/ 99.4%

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

   14. associate-/l/ 99.4%

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

   15. associate-/r/ 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right)} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 56.5%

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

   2. pow2 56.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right)}\right)}^{2}} \]

5. Applied egg-rr 56.5%

   \[\leadsto \color{blue}{{\left(\sqrt{\left(\sqrt{x} \cdot 3\right) \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right)}^{2}} \]
6. Step-by-step derivation

   1. unpow2 56.5%

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

   2. sqrt-unprod 49.1%

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

   3. *-commutative 49.1%

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

   4. *-commutative 49.1%

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

   5. swap-sqr 29.0%

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

   6. pow2 29.0%

      \[\leadsto \sqrt{\color{blue}{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2}} \cdot \left(\left(\sqrt{x} \cdot 3\right) \cdot \left(\sqrt{x} \cdot 3\right)\right)} \]

   7. swap-sqr 29.0%

      \[\leadsto \sqrt{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2} \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(3 \cdot 3\right)\right)}} \]

   8. add-sqr-sqrt 29.0%

      \[\leadsto \sqrt{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2} \cdot \left(\color{blue}{x} \cdot \left(3 \cdot 3\right)\right)} \]

   9. metadata-eval 29.0%

      \[\leadsto \sqrt{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2} \cdot \left(x \cdot \color{blue}{9}\right)} \]

7. Applied egg-rr 29.0%

   \[\leadsto \color{blue}{\sqrt{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2} \cdot \left(x \cdot 9\right)}} \]
8. Step-by-step derivation

   1. sqrt-prod 31.6%

      \[\leadsto \color{blue}{\sqrt{{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}^{2}} \cdot \sqrt{x \cdot 9}} \]

   2. unpow2 31.6%

      \[\leadsto \sqrt{\color{blue}{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)}} \cdot \sqrt{x \cdot 9} \]

   3. sqrt-prod 56.5%

      \[\leadsto \color{blue}{\left(\sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)} \cdot \sqrt{y + \left(\frac{0.1111111111111111}{x} + -1\right)}\right)} \cdot \sqrt{x \cdot 9} \]

   4. add-sqr-sqrt 99.5%

      \[\leadsto \color{blue}{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)} \cdot \sqrt{x \cdot 9} \]

   5. add-sqr-sqrt 99.0%

      \[\leadsto \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \color{blue}{\left(\sqrt{\sqrt{x \cdot 9}} \cdot \sqrt{\sqrt{x \cdot 9}}\right)} \]

   6. associate-*r* 99.1%

      \[\leadsto \color{blue}{\left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \sqrt{\sqrt{x \cdot 9}}\right) \cdot \sqrt{\sqrt{x \cdot 9}}} \]

   7. pow1/2 99.1%

      \[\leadsto \left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \sqrt{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}}\right) \cdot \sqrt{\sqrt{x \cdot 9}} \]

   8. sqrt-pow1 99.2%

      \[\leadsto \left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \color{blue}{{\left(x \cdot 9\right)}^{\left(\frac{0.5}{2}\right)}}\right) \cdot \sqrt{\sqrt{x \cdot 9}} \]

   9. metadata-eval 99.2%

      \[\leadsto \left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot {\left(x \cdot 9\right)}^{\color{blue}{0.25}}\right) \cdot \sqrt{\sqrt{x \cdot 9}} \]

   10. pow1/2 99.2%

       \[\leadsto \left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot {\left(x \cdot 9\right)}^{0.25}\right) \cdot \sqrt{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   11. sqrt-pow1 99.1%

       \[\leadsto \left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot {\left(x \cdot 9\right)}^{0.25}\right) \cdot \color{blue}{{\left(x \cdot 9\right)}^{\left(\frac{0.5}{2}\right)}} \]

   12. metadata-eval 99.1%

       \[\leadsto \left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot {\left(x \cdot 9\right)}^{0.25}\right) \cdot {\left(x \cdot 9\right)}^{\color{blue}{0.25}} \]

9. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot {\left(x \cdot 9\right)}^{0.25}\right) \cdot {\left(x \cdot 9\right)}^{0.25}} \]
10. Step-by-step derivation

    1. associate-*l* 99.0%

       \[\leadsto \color{blue}{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left({\left(x \cdot 9\right)}^{0.25} \cdot {\left(x \cdot 9\right)}^{0.25}\right)} \]

    2. pow-sqr 99.5%

       \[\leadsto \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \color{blue}{{\left(x \cdot 9\right)}^{\left(2 \cdot 0.25\right)}} \]

    3. metadata-eval 99.5%

       \[\leadsto \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot {\left(x \cdot 9\right)}^{\color{blue}{0.5}} \]

    4. unpow1/2 99.5%

       \[\leadsto \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \color{blue}{\sqrt{x \cdot 9}} \]

11. Simplified 99.5%

    \[\leadsto \color{blue}{\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \sqrt{x \cdot 9}} \]

12. Final simplification 99.5%

    \[\leadsto \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \sqrt{x \cdot 9} \]

Alternative 8: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1050 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1050.0) (not (<= y 1.0)))
   (* 3.0 (* (sqrt x) y))
   (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1050.0) || !(y <= 1.0)) {
		tmp = 3.0 * (sqrt(x) * y);
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1050.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1050.0) || !(y <= 1.0)) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1050.0) or not (y <= 1.0):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1050.0) || !(y <= 1.0))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1050.0) || ~((y <= 1.0)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1050.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1050 or 1 < y

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 73.0%

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

   if -1050 < y < 1

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-rgt-in 99.4%

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

      4. remove-double-neg 99.4%

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

      5. distribute-lft-neg-in 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. mul-1-neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-/r* 99.4%

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

      11. distribute-neg-frac 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r/ 99.4%

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

      14. associate-/l/ 99.4%

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

      15. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 49.7%

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

   5. Taylor expanded in y around 0 48.8%

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

      1. *-commutative 48.8%

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

   7. Simplified 48.8%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1050 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 9: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1050:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1050.0)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 1.0) (* (sqrt x) -3.0) (* (sqrt x) (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1050.0) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 1.0) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1050.0d0)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 1.0d0) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1050.0) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 1.0) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1050.0:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 1.0:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1050.0)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 1.0)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1050.0)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 1.0)
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1050.0], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1050

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 72.8%

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

   if -1050 < y < 1

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-rgt-in 99.4%

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

      4. remove-double-neg 99.4%

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

      5. distribute-lft-neg-in 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. mul-1-neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-/r* 99.4%

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

      11. distribute-neg-frac 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r/ 99.4%

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

      14. associate-/l/ 99.4%

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

      15. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 49.7%

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

   5. Taylor expanded in y around 0 48.8%

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

      1. *-commutative 48.8%

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

   7. Simplified 48.8%

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

   if 1 < y

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 73.1%

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

      1. associate-*r* 73.2%

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

      2. *-commutative 73.2%

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

   6. Simplified 73.2%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1050:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]

Alternative 10: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.5e-5)
   (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
   (* (sqrt x) (- (* 3.0 y) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.5e-5) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 5.5d-5) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.5e-5) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.5e-5:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.5e-5)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) - 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.5e-5)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.5e-5], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5000000000000002e-5

   1. Initial program 99.2%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 79.1%

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

      1. *-commutative 79.1%

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

      2. sub-neg 79.1%

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

      3. associate-*r/ 79.3%

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

      4. metadata-eval 79.3%

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

      5. metadata-eval 79.3%

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

   6. Simplified 79.3%

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

   if 5.5000000000000002e-5 < x

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]

Alternative 11: 3.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (* x 9.0)))

C

double code(double x, double y) {
	return sqrt((x * 9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((x * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((x * 9.0));
}

Python

def code(x, y):
	return math.sqrt((x * 9.0))

Julia

function code(x, y)
	return sqrt(Float64(x * 9.0))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((x * 9.0));
end

Wolfram

code[x_, y_] := N[Sqrt[N[(x * 9.0), $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. Step-by-step derivation

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. distribute-rgt-in 99.4%

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

   4. remove-double-neg 99.4%

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

   5. distribute-lft-neg-in 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. mul-1-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. *-commutative 99.4%

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

   10. associate-/r* 99.4%

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

   11. distribute-neg-frac 99.4%

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

   12. *-commutative 99.4%

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

   13. associate-/r/ 99.4%

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

   14. associate-/l/ 99.4%

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

   15. associate-/r/ 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around inf 60.6%

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

5. Taylor expanded in y around 0 28.4%

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

   1. *-commutative 28.4%

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

7. Simplified 28.4%

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

   1. expm1-log1p-u 0.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x} \cdot -3\right)\right)} \]

   2. expm1-udef 1.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{x} \cdot -3\right)} - 1} \]

   3. add-sqr-sqrt 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\sqrt{x} \cdot -3} \cdot \sqrt{\sqrt{x} \cdot -3}}\right)} - 1 \]

   4. sqrt-unprod 2.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}}\right)} - 1 \]

   5. swap-sqr 2.5%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(-3 \cdot -3\right)}}\right)} - 1 \]

   6. add-sqr-sqrt 2.5%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x} \cdot \left(-3 \cdot -3\right)}\right)} - 1 \]

   7. metadata-eval 2.5%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{x \cdot \color{blue}{9}}\right)} - 1 \]

9. Applied egg-rr 2.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 3.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \]

    2. expm1-log1p 3.3%

       \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \]

11. Simplified 3.3%

    \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \]

12. Final simplification 3.3%

    \[\leadsto \sqrt{x \cdot 9} \]

Alternative 12: 26.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. distribute-rgt-in 99.4%

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

   4. remove-double-neg 99.4%

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

   5. distribute-lft-neg-in 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. mul-1-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. *-commutative 99.4%

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

   10. associate-/r* 99.4%

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

   11. distribute-neg-frac 99.4%

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

   12. *-commutative 99.4%

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

   13. associate-/r/ 99.4%

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

   14. associate-/l/ 99.4%

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

   15. associate-/r/ 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around inf 60.6%

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

5. Taylor expanded in y around 0 28.4%

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

   1. *-commutative 28.4%

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

7. Simplified 28.4%

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

8. Final simplification 28.4%

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

Developer target: 99.4% accurate, 0.5× 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 2023200 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))