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

Percentage Accurate: 99.4% → 99.5%

Time: 11.3s

Alternatives: 12

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.3333333333333333d0 / sqrt(x)) + ((sqrt(x) * 3.0d0) * (y + (-1.0d0)))
end function

Java

public static double code(double x, double y) {
	return (0.3333333333333333 / Math.sqrt(x)) + ((Math.sqrt(x) * 3.0) * (y + -1.0));
}

Python

def code(x, y):
	return (0.3333333333333333 / math.sqrt(x)) + ((math.sqrt(x) * 3.0) * (y + -1.0))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (0.3333333333333333 / sqrt(x)) + ((sqrt(x) * 3.0) * (y + -1.0));
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y + -1.0), $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. associate--l+ 99.4%

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

   2. associate-/r* 99.5%

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

3. Simplified 99.5%

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

   1. associate-+r- 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-/r* 99.4%

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

   4. inv-pow 99.4%

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

   5. *-commutative 99.4%

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

   6. unpow-prod-down 99.4%

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

   7. metadata-eval 99.4%

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

   8. inv-pow 99.4%

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

   9. div-inv 99.4%

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

   10. associate-+r- 99.4%

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

   11. distribute-rgt-in 99.4%

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

   12. sub-neg 99.4%

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

   13. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. clear-num 99.4%

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

   3. un-div-inv 99.4%

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

   4. div-inv 99.4%

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

   5. metadata-eval 99.4%

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

7. Applied egg-rr 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. *-commutative 99.4%

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

   3. metadata-eval 99.4%

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

   4. add-sqr-sqrt 99.4%

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

   5. swap-sqr 99.5%

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

   6. times-frac 99.5%

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

   7. metadata-eval 99.5%

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

   8. *-commutative 99.5%

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

   9. metadata-eval 99.5%

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

   10. sqrt-prod 99.6%

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

   11. metadata-eval 99.6%

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

   12. div-inv 99.6%

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

   13. sqrt-div 99.6%

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

   14. clear-num 99.6%

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

   15. sqrt-div 99.6%

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

   16. metadata-eval 99.6%

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

9. Applied egg-rr 99.6%

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

    1. *-inverses 99.6%

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

    2. *-rgt-identity 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 10^{+63} \lor \neg \left(x \leq 4.2 \cdot 10^{+169}\right) \land \left(x \leq 4 \cdot 10^{+185} \lor \neg \left(x \leq 3 \cdot 10^{+212}\right) \land x \leq 7.2 \cdot 10^{+233}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.5e-24)
   (* (sqrt x) (/ 0.3333333333333333 x))
   (if (or (<= x 1e+63)
           (and (not (<= x 4.2e+169))
                (or (<= x 4e+185) (and (not (<= x 3e+212)) (<= x 7.2e+233)))))
     (* 3.0 (* (sqrt x) y))
     (* (sqrt x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.5e-24) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else if ((x <= 1e+63) || (!(x <= 4.2e+169) && ((x <= 4e+185) || (!(x <= 3e+212) && (x <= 7.2e+233))))) {
		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 (x <= 1.5d-24) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else if ((x <= 1d+63) .or. (.not. (x <= 4.2d+169)) .and. (x <= 4d+185) .or. (.not. (x <= 3d+212)) .and. (x <= 7.2d+233)) 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 (x <= 1.5e-24) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else if ((x <= 1e+63) || (!(x <= 4.2e+169) && ((x <= 4e+185) || (!(x <= 3e+212) && (x <= 7.2e+233))))) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.5e-24:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	elif (x <= 1e+63) or (not (x <= 4.2e+169) and ((x <= 4e+185) or (not (x <= 3e+212) and (x <= 7.2e+233)))):
		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 (x <= 1.5e-24)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	elseif ((x <= 1e+63) || (!(x <= 4.2e+169) && ((x <= 4e+185) || (!(x <= 3e+212) && (x <= 7.2e+233)))))
		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 (x <= 1.5e-24)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	elseif ((x <= 1e+63) || (~((x <= 4.2e+169)) && ((x <= 4e+185) || (~((x <= 3e+212)) && (x <= 7.2e+233)))))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.5e-24], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 1e+63], And[N[Not[LessEqual[x, 4.2e+169]], $MachinePrecision], Or[LessEqual[x, 4e+185], And[N[Not[LessEqual[x, 3e+212]], $MachinePrecision], LessEqual[x, 7.2e+233]]]]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.49999999999999998e-24

   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 99.3%

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

   4. Taylor expanded in x around 0 71.9%

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

   if 1.49999999999999998e-24 < x < 1.00000000000000006e63 or 4.2000000000000002e169 < x < 3.9999999999999999e185 or 3e212 < x < 7.1999999999999996e233

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 64.5%

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

   if 1.00000000000000006e63 < x < 4.2000000000000002e169 or 3.9999999999999999e185 < x < 3e212 or 7.1999999999999996e233 < 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 99.6%

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

   5. Taylor expanded in y around 0 66.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 10^{+63} \lor \neg \left(x \leq 4.2 \cdot 10^{+169}\right) \land \left(x \leq 4 \cdot 10^{+185} \lor \neg \left(x \leq 3 \cdot 10^{+212}\right) \land x \leq 7.2 \cdot 10^{+233}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 3: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ t_1 := \sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{if}\;x \leq 2.5 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+212} \lor \neg \left(x \leq 2.2 \cdot 10^{+233}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0)) (t_1 (* (sqrt x) (* y 3.0))))
   (if (<= x 2.5e-24)
     (* (sqrt x) (/ 0.3333333333333333 x))
     (if (<= x 1.8e+63)
       t_1
       (if (<= x 2.9e+167)
         t_0
         (if (<= x 1.35e+182)
           t_1
           (if (or (<= x 5.8e+212) (not (<= x 2.2e+233)))
             t_0
             (* 3.0 (* (sqrt x) y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double t_1 = sqrt(x) * (y * 3.0);
	double tmp;
	if (x <= 2.5e-24) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else if (x <= 1.8e+63) {
		tmp = t_1;
	} else if (x <= 2.9e+167) {
		tmp = t_0;
	} else if (x <= 1.35e+182) {
		tmp = t_1;
	} else if ((x <= 5.8e+212) || !(x <= 2.2e+233)) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (sqrt(x) * y);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    t_1 = sqrt(x) * (y * 3.0d0)
    if (x <= 2.5d-24) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else if (x <= 1.8d+63) then
        tmp = t_1
    else if (x <= 2.9d+167) then
        tmp = t_0
    else if (x <= 1.35d+182) then
        tmp = t_1
    else if ((x <= 5.8d+212) .or. (.not. (x <= 2.2d+233))) then
        tmp = t_0
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double t_1 = Math.sqrt(x) * (y * 3.0);
	double tmp;
	if (x <= 2.5e-24) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else if (x <= 1.8e+63) {
		tmp = t_1;
	} else if (x <= 2.9e+167) {
		tmp = t_0;
	} else if (x <= 1.35e+182) {
		tmp = t_1;
	} else if ((x <= 5.8e+212) || !(x <= 2.2e+233)) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	t_1 = math.sqrt(x) * (y * 3.0)
	tmp = 0
	if x <= 2.5e-24:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	elif x <= 1.8e+63:
		tmp = t_1
	elif x <= 2.9e+167:
		tmp = t_0
	elif x <= 1.35e+182:
		tmp = t_1
	elif (x <= 5.8e+212) or not (x <= 2.2e+233):
		tmp = t_0
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	t_1 = Float64(sqrt(x) * Float64(y * 3.0))
	tmp = 0.0
	if (x <= 2.5e-24)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	elseif (x <= 1.8e+63)
		tmp = t_1;
	elseif (x <= 2.9e+167)
		tmp = t_0;
	elseif (x <= 1.35e+182)
		tmp = t_1;
	elseif ((x <= 5.8e+212) || !(x <= 2.2e+233))
		tmp = t_0;
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	t_1 = sqrt(x) * (y * 3.0);
	tmp = 0.0;
	if (x <= 2.5e-24)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	elseif (x <= 1.8e+63)
		tmp = t_1;
	elseif (x <= 2.9e+167)
		tmp = t_0;
	elseif (x <= 1.35e+182)
		tmp = t_1;
	elseif ((x <= 5.8e+212) || ~((x <= 2.2e+233)))
		tmp = t_0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.5e-24], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e+63], t$95$1, If[LessEqual[x, 2.9e+167], t$95$0, If[LessEqual[x, 1.35e+182], t$95$1, If[Or[LessEqual[x, 5.8e+212], N[Not[LessEqual[x, 2.2e+233]], $MachinePrecision]], t$95$0, N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.4999999999999999e-24

   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 99.3%

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

   4. Taylor expanded in x around 0 71.9%

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

   if 2.4999999999999999e-24 < x < 1.79999999999999999e63 or 2.89999999999999975e167 < x < 1.3500000000000001e182

   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. associate--l+ 99.6%

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

      2. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 60.7%

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

      1. associate-*r* 60.8%

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

   6. Simplified 60.8%

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

   if 1.79999999999999999e63 < x < 2.89999999999999975e167 or 1.3500000000000001e182 < x < 5.7999999999999997e212 or 2.19999999999999999e233 < 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 99.6%

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

   5. Taylor expanded in y around 0 66.8%

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

   if 5.7999999999999997e212 < x < 2.19999999999999999e233

   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. associate--l+ 99.4%

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

      2. associate-/r* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 81.1%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+182}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+212} \lor \neg \left(x \leq 2.2 \cdot 10^{+233}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 4: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ t_1 := y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{if}\;x \leq 1.62 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+212} \lor \neg \left(x \leq 1.02 \cdot 10^{+235}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0)) (t_1 (* y (* (sqrt x) 3.0))))
   (if (<= x 1.62e-24)
     (* (sqrt x) (/ 0.3333333333333333 x))
     (if (<= x 4.2e+63)
       t_1
       (if (<= x 9.5e+166)
         t_0
         (if (<= x 5.6e+181)
           t_1
           (if (or (<= x 3.9e+212) (not (<= x 1.02e+235)))
             t_0
             (* 3.0 (* (sqrt x) y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double t_1 = y * (sqrt(x) * 3.0);
	double tmp;
	if (x <= 1.62e-24) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else if (x <= 4.2e+63) {
		tmp = t_1;
	} else if (x <= 9.5e+166) {
		tmp = t_0;
	} else if (x <= 5.6e+181) {
		tmp = t_1;
	} else if ((x <= 3.9e+212) || !(x <= 1.02e+235)) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (sqrt(x) * y);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    t_1 = y * (sqrt(x) * 3.0d0)
    if (x <= 1.62d-24) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else if (x <= 4.2d+63) then
        tmp = t_1
    else if (x <= 9.5d+166) then
        tmp = t_0
    else if (x <= 5.6d+181) then
        tmp = t_1
    else if ((x <= 3.9d+212) .or. (.not. (x <= 1.02d+235))) then
        tmp = t_0
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double t_1 = y * (Math.sqrt(x) * 3.0);
	double tmp;
	if (x <= 1.62e-24) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else if (x <= 4.2e+63) {
		tmp = t_1;
	} else if (x <= 9.5e+166) {
		tmp = t_0;
	} else if (x <= 5.6e+181) {
		tmp = t_1;
	} else if ((x <= 3.9e+212) || !(x <= 1.02e+235)) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	t_1 = y * (math.sqrt(x) * 3.0)
	tmp = 0
	if x <= 1.62e-24:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	elif x <= 4.2e+63:
		tmp = t_1
	elif x <= 9.5e+166:
		tmp = t_0
	elif x <= 5.6e+181:
		tmp = t_1
	elif (x <= 3.9e+212) or not (x <= 1.02e+235):
		tmp = t_0
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	t_1 = Float64(y * Float64(sqrt(x) * 3.0))
	tmp = 0.0
	if (x <= 1.62e-24)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	elseif (x <= 4.2e+63)
		tmp = t_1;
	elseif (x <= 9.5e+166)
		tmp = t_0;
	elseif (x <= 5.6e+181)
		tmp = t_1;
	elseif ((x <= 3.9e+212) || !(x <= 1.02e+235))
		tmp = t_0;
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	t_1 = y * (sqrt(x) * 3.0);
	tmp = 0.0;
	if (x <= 1.62e-24)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	elseif (x <= 4.2e+63)
		tmp = t_1;
	elseif (x <= 9.5e+166)
		tmp = t_0;
	elseif (x <= 5.6e+181)
		tmp = t_1;
	elseif ((x <= 3.9e+212) || ~((x <= 1.02e+235)))
		tmp = t_0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.62e-24], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e+63], t$95$1, If[LessEqual[x, 9.5e+166], t$95$0, If[LessEqual[x, 5.6e+181], t$95$1, If[Or[LessEqual[x, 3.9e+212], N[Not[LessEqual[x, 1.02e+235]], $MachinePrecision]], t$95$0, N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.62e-24

   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 99.3%

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

   4. Taylor expanded in x around 0 71.9%

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

   if 1.62e-24 < x < 4.2000000000000004e63 or 9.49999999999999984e166 < x < 5.59999999999999968e181

   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. associate--l+ 99.6%

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

      2. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 60.9%

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

   if 4.2000000000000004e63 < x < 9.49999999999999984e166 or 5.59999999999999968e181 < x < 3.9000000000000001e212 or 1.02e235 < 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 99.6%

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

   5. Taylor expanded in y around 0 66.8%

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

   if 3.9000000000000001e212 < x < 1.02e235

   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. associate--l+ 99.4%

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

      2. associate-/r* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 81.1%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.62 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+63}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+166}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+181}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+212} \lor \neg \left(x \leq 1.02 \cdot 10^{+235}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate--l+ 99.4%

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

   2. associate-/r* 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+50} \lor \neg \left(y \leq 2.3 \cdot 10^{+15}\right):\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.2e+50) (not (<= y 2.3e+15)))
   (* y (* (sqrt x) 3.0))
   (* (sqrt x) (- -3.0 (/ -0.3333333333333333 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.2e+50) || !(y <= 2.3e+15)) {
		tmp = y * (sqrt(x) * 3.0);
	} else {
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3.2d+50)) .or. (.not. (y <= 2.3d+15))) then
        tmp = y * (sqrt(x) * 3.0d0)
    else
        tmp = sqrt(x) * ((-3.0d0) - ((-0.3333333333333333d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.2e+50) || !(y <= 2.3e+15)) {
		tmp = y * (Math.sqrt(x) * 3.0);
	} else {
		tmp = Math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.2e+50) or not (y <= 2.3e+15):
		tmp = y * (math.sqrt(x) * 3.0)
	else:
		tmp = math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.2e+50) || !(y <= 2.3e+15))
		tmp = Float64(y * Float64(sqrt(x) * 3.0));
	else
		tmp = Float64(sqrt(x) * Float64(-3.0 - Float64(-0.3333333333333333 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.2e+50) || ~((y <= 2.3e+15)))
		tmp = y * (sqrt(x) * 3.0);
	else
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.2e+50], N[Not[LessEqual[y, 2.3e+15]], $MachinePrecision]], N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 - N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999983e50 or 2.3e15 < y

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 77.8%

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

   if -3.19999999999999983e50 < y < 2.3e15

   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. associate--l+ 99.4%

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

      2. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 92.1%

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

      1. associate-*r* 92.1%

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

      2. *-commutative 92.1%

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

      3. sub-neg 92.1%

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

      4. associate-*r/ 92.2%

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

      5. metadata-eval 92.2%

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

      6. metadata-eval 92.2%

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

      7. distribute-rgt-in 92.2%

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

      8. associate-*l/ 92.2%

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

      9. metadata-eval 92.2%

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

      10. metadata-eval 92.2%

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

      11. +-commutative 92.2%

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

      12. metadata-eval 92.2%

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

      13. distribute-neg-frac 92.2%

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

      14. unsub-neg 92.2%

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

   6. Simplified 92.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+50} \lor \neg \left(y \leq 2.3 \cdot 10^{+15}\right):\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \end{array} \]

Alternative 7: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(N[(0.1111111111111111 / x), $MachinePrecision] + N[(y + -1.0), $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. Final simplification 99.4%

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

Alternative 8: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(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 <= (-1.0d0)) .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 <= -1.0) || !(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 <= -1.0) 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 <= -1.0) || !(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 <= -1.0) || ~((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, -1.0], 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 < -1 or 1 < y

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 69.9%

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

   if -1 < y < 1

   1. Initial program 99.4%

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

   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 inf 56.2%

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

   5. Taylor expanded in y around 0 53.4%

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

4. Final simplification 61.6%

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

Alternative 9: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 80:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 80.0)
   (* (sqrt x) (- -3.0 (/ -0.3333333333333333 x)))
   (* (sqrt x) (- (* y 3.0) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 80.0) {
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * ((y * 3.0) - 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 <= 80.0d0) then
        tmp = sqrt(x) * ((-3.0d0) - ((-0.3333333333333333d0) / x))
    else
        tmp = sqrt(x) * ((y * 3.0d0) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 80.0) {
		tmp = Math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt(x) * ((y * 3.0) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 80.0:
		tmp = math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x))
	else:
		tmp = math.sqrt(x) * ((y * 3.0) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 80.0)
		tmp = Float64(sqrt(x) * Float64(-3.0 - Float64(-0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(y * 3.0) - 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 80.0)
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	else
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 80.0], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 - N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y * 3.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 80

   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. associate--l+ 99.2%

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

      2. associate-/r* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. *-commutative 71.0%

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

      3. sub-neg 71.0%

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

      4. associate-*r/ 71.1%

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

      5. metadata-eval 71.1%

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

      6. metadata-eval 71.1%

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

      7. distribute-rgt-in 71.1%

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

      8. associate-*l/ 71.1%

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

      9. metadata-eval 71.1%

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

      10. metadata-eval 71.1%

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

      11. +-commutative 71.1%

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

      12. metadata-eval 71.1%

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

      13. distribute-neg-frac 71.1%

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

      14. unsub-neg 71.1%

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

   6. Simplified 71.1%

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

   if 80 < 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 99.5%

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

   4. Taylor expanded in x around inf 98.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 80:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \end{array} \]

Alternative 10: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 80:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 80.0)
   (* (sqrt x) (- -3.0 (/ -0.3333333333333333 x)))
   (* (* (sqrt x) 3.0) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 80.0) {
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	} else {
		tmp = (sqrt(x) * 3.0) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 80.0d0) then
        tmp = sqrt(x) * ((-3.0d0) - ((-0.3333333333333333d0) / x))
    else
        tmp = (sqrt(x) * 3.0d0) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 80.0) {
		tmp = Math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	} else {
		tmp = (Math.sqrt(x) * 3.0) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 80.0:
		tmp = math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x))
	else:
		tmp = (math.sqrt(x) * 3.0) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 80.0)
		tmp = Float64(sqrt(x) * Float64(-3.0 - Float64(-0.3333333333333333 / x)));
	else
		tmp = Float64(Float64(sqrt(x) * 3.0) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 80.0)
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	else
		tmp = (sqrt(x) * 3.0) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 80.0], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 - N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 80

   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. associate--l+ 99.2%

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

      2. associate-/r* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. *-commutative 71.0%

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

      3. sub-neg 71.0%

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

      4. associate-*r/ 71.1%

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

      5. metadata-eval 71.1%

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

      6. metadata-eval 71.1%

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

      7. distribute-rgt-in 71.1%

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

      8. associate-*l/ 71.1%

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

      9. metadata-eval 71.1%

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

      10. metadata-eval 71.1%

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

      11. +-commutative 71.1%

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

      12. metadata-eval 71.1%

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

      13. distribute-neg-frac 71.1%

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

      14. unsub-neg 71.1%

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

   6. Simplified 71.1%

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

   if 80 < 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. associate--l+ 99.6%

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

      2. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 80:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 11: 3.3% 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%

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

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 inf 63.7%

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

5. Taylor expanded in y around 0 28.1%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 3.1%

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

   3. swap-sqr 3.1%

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

   4. metadata-eval 3.1%

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

   5. add-sqr-sqrt 3.1%

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

   6. *-commutative 3.1%

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

   7. pow1/2 3.1%

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

7. Applied egg-rr 3.1%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \]
8. Step-by-step derivation

   1. unpow1/2 3.1%

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

9. Simplified 3.1%

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

10. Final simplification 3.1%

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

Alternative 12: 26.0% 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%

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

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 inf 63.7%

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

5. Taylor expanded in y around 0 28.1%

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

6. Final simplification 28.1%

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

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 2023196 
(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)))