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

Percentage Accurate: 99.4% → 99.4%

Time: 9.6s

Alternatives: 12

Speedup: N/A×

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, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return sqrt((x * 9.0)) * ((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 = sqrt((x * 9.0d0)) * ((y + (1.0d0 / (x * 9.0d0))) + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. sqrt-prod 99.6%

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

   4. pow1/2 99.6%

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

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
5. Step-by-step derivation

   1. unpow1/2 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+38)
   (* (sqrt x) (* y 3.0))
   (if (<= y 1.65e+17)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (* (sqrt (* x 9.0)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+38) {
		tmp = sqrt(x) * (y * 3.0);
	} else if (y <= 1.65e+17) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = sqrt((x * 9.0)) * 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 <= (-3.8d+38)) then
        tmp = sqrt(x) * (y * 3.0d0)
    else if (y <= 1.65d+17) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = sqrt((x * 9.0d0)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+38) {
		tmp = Math.sqrt(x) * (y * 3.0);
	} else if (y <= 1.65e+17) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt((x * 9.0)) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+38:
		tmp = math.sqrt(x) * (y * 3.0)
	elif y <= 1.65e+17:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = math.sqrt((x * 9.0)) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+38)
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	elseif (y <= 1.65e+17)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.8e+38)
		tmp = sqrt(x) * (y * 3.0);
	elseif (y <= 1.65e+17)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = sqrt((x * 9.0)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+38], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e+17], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7999999999999998e38

   1. Initial program 99.3%

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

      1. associate-*l* 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 77.7%

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

      1. *-commutative 77.7%

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

      2. associate-*l* 77.7%

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

      3. *-commutative 77.7%

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

   7. Simplified 77.7%

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

   if -3.7999999999999998e38 < y < 1.65e17

   1. Initial program 99.3%

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

      1. associate-*l* 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around 0 93.5%

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

      1. associate-*r* 93.5%

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

      2. sub-neg 93.5%

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

      3. associate-*r/ 93.6%

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

      4. metadata-eval 93.6%

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

      5. metadata-eval 93.6%

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

      6. distribute-rgt-in 93.6%

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

      7. associate-*r* 93.6%

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

      8. associate-*l/ 93.6%

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

      9. metadata-eval 93.6%

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

      10. associate-*r* 93.6%

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

      11. metadata-eval 93.6%

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

      12. distribute-rgt-in 93.6%

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

   7. Simplified 93.6%

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

   if 1.65e17 < y

   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* 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/r* 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 75.8%

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

      1. expm1-log1p-u 70.2%

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

      2. expm1-udef 70.0%

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

      3. *-commutative 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-*l* 70.0%

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

      6. metadata-eval 70.0%

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

      7. sqrt-prod 70.0%

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

   7. Applied egg-rr 70.0%

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

      1. expm1-def 70.2%

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

      2. expm1-log1p 75.9%

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

   9. Simplified 75.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;\sqrt{x} \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.032)
   (* (sqrt x) (- (* 0.3333333333333333 (/ 1.0 x)) 3.0))
   (* (sqrt x) (- (* y 3.0) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.032) {
		tmp = sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	} 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 <= 0.032d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 * (1.0d0 / x)) - 3.0d0)
    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 <= 0.032) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	} else {
		tmp = Math.sqrt(x) * ((y * 3.0) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.032:
		tmp = math.sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0)
	else:
		tmp = math.sqrt(x) * ((y * 3.0) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.032)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 3.0));
	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 <= 0.032)
		tmp = sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	else
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.032], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 3.0), $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 < 0.032000000000000001

   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%

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

      2. associate-*l* 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-lft-in 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

      8. *-commutative 99.3%

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

      9. associate-/r* 99.3%

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

      10. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 76.4%

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

   if 0.032000000000000001 < 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%

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

      2. associate-*l* 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/r* 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;\sqrt{x} \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 3.0 * (sqrt(x) * (y + ((0.1111111111111111 / x) + -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 + ((0.1111111111111111d0 / x) + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $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* 99.3%

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

   2. associate--l+ 99.3%

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

   3. sub-neg 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return sqrt(x) * (-3.0 + ((y + (0.1111111111111111 / 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 + (0.1111111111111111d0 / x)) * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l* 99.4%

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

   3. sub-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-lft-in 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-/r* 99.4%

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

   10. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

   \[\leadsto \sqrt{x} \cdot \left(-3 + \left(y + \frac{0.1111111111111111}{x}\right) \cdot 3\right) \]
6. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((x * 9.0d0)) * ((y + (0.1111111111111111d0 / x)) + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. sqrt-prod 99.6%

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

   4. pow1/2 99.6%

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

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
5. Step-by-step derivation

   1. unpow1/2 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around 0 99.5%

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

8. Final simplification 99.5%

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

Alternative 7: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.016:\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot \frac{1}{x} - 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.016)
   (sqrt (- (* 0.1111111111111111 (/ 1.0 x)) 2.0))
   (* (sqrt (* x 9.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.016) {
		tmp = sqrt(((0.1111111111111111 * (1.0 / x)) - 2.0));
	} else {
		tmp = sqrt((x * 9.0)) * 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 <= 0.016d0) then
        tmp = sqrt(((0.1111111111111111d0 * (1.0d0 / x)) - 2.0d0))
    else
        tmp = sqrt((x * 9.0d0)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.016) {
		tmp = Math.sqrt(((0.1111111111111111 * (1.0 / x)) - 2.0));
	} else {
		tmp = Math.sqrt((x * 9.0)) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.016:
		tmp = math.sqrt(((0.1111111111111111 * (1.0 / x)) - 2.0))
	else:
		tmp = math.sqrt((x * 9.0)) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.016)
		tmp = sqrt(Float64(Float64(0.1111111111111111 * Float64(1.0 / x)) - 2.0));
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.016)
		tmp = sqrt(((0.1111111111111111 * (1.0 / x)) - 2.0));
	else
		tmp = sqrt((x * 9.0)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.016], N[Sqrt[N[(N[(0.1111111111111111 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.016

   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%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 76.3%

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

      1. sub-neg 76.3%

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

      2. associate-*r/ 76.3%

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

      3. metadata-eval 76.3%

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

      4. metadata-eval 76.3%

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

   7. Simplified 76.3%

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

      1. add-sqr-sqrt 76.0%

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

      2. sqrt-unprod 76.3%

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

      3. associate-*r* 76.3%

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

      4. associate-*r* 76.3%

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

      5. swap-sqr 34.9%

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

      6. swap-sqr 35.0%

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

      7. metadata-eval 35.0%

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

      8. add-sqr-sqrt 35.0%

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

      9. *-commutative 35.0%

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

      10. pow2 35.0%

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

   9. Applied egg-rr 35.0%

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

       1. associate-*l* 35.0%

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

   11. Simplified 35.0%

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

   12. Taylor expanded in x around 0 76.0%

       \[\leadsto \sqrt{\color{blue}{0.1111111111111111 \cdot \frac{1}{x} - 2}} \]

   if 0.016 < 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* 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 48.3%

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

      1. expm1-log1p-u 22.8%

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

      2. expm1-udef 22.7%

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

      3. *-commutative 22.7%

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

      4. *-commutative 22.7%

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

      5. associate-*l* 22.7%

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

      6. metadata-eval 22.7%

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

      7. sqrt-prod 22.7%

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

   7. Applied egg-rr 22.7%

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

      1. expm1-def 22.8%

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

      2. expm1-log1p 48.3%

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

   9. Simplified 48.3%

      \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.016:\\ \;\;\;\;\sqrt{0.1111111111111111 \cdot \frac{1}{x} - 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.044:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.044)
   (* (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 <= 0.044) {
		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 <= 0.044d0) 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 <= 0.044) {
		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 <= 0.044:
		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 <= 0.044)
		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 <= 0.044)
		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, 0.044], 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 < 0.043999999999999997

   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%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 76.3%

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

      1. associate-*r* 76.3%

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

      2. sub-neg 76.3%

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

      3. associate-*r/ 76.3%

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

      4. metadata-eval 76.3%

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

      5. metadata-eval 76.3%

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

      6. distribute-rgt-in 76.3%

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

      7. associate-*r* 76.3%

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

      8. associate-*l/ 76.4%

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

      9. metadata-eval 76.4%

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

      10. associate-*r* 76.4%

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

      11. metadata-eval 76.4%

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

      12. distribute-rgt-in 76.4%

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

   7. Simplified 76.4%

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

   if 0.043999999999999997 < 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%

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

      2. associate-*l* 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/r* 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.044:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.016:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.016)
   (sqrt (+ (/ 0.1111111111111111 x) -2.0))
   (* 3.0 (* y (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.016) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = 3.0 * (y * sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.016d0) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    else
        tmp = 3.0d0 * (y * sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.016) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = 3.0 * (y * Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.016:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	else:
		tmp = 3.0 * (y * math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.016)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	else
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.016)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	else
		tmp = 3.0 * (y * sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.016], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.016

   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%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 76.3%

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

      1. sub-neg 76.3%

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

      2. associate-*r/ 76.3%

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

      3. metadata-eval 76.3%

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

      4. metadata-eval 76.3%

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

   7. Simplified 76.3%

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

      1. add-sqr-sqrt 76.0%

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

      2. sqrt-unprod 76.3%

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

      3. associate-*r* 76.3%

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

      4. associate-*r* 76.3%

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

      5. swap-sqr 34.9%

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

      6. swap-sqr 35.0%

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

      7. metadata-eval 35.0%

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

      8. add-sqr-sqrt 35.0%

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

      9. *-commutative 35.0%

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

      10. pow2 35.0%

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

   9. Applied egg-rr 35.0%

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

       1. associate-*l* 35.0%

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

   11. Simplified 35.0%

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

   12. Taylor expanded in x around 0 76.0%

       \[\leadsto \sqrt{\color{blue}{0.1111111111111111 \cdot \frac{1}{x} - 2}} \]
   13. Step-by-step derivation

       1. sub-neg 76.0%

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

       2. associate-*r/ 75.9%

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

       3. metadata-eval 75.9%

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

       4. metadata-eval 75.9%

          \[\leadsto \sqrt{\frac{0.1111111111111111}{x} + \color{blue}{-2}} \]

   14. Simplified 75.9%

       \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x} + -2}} \]

   if 0.016 < 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* 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 48.3%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.016:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.016:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.016)
   (sqrt (+ (/ 0.1111111111111111 x) -2.0))
   (* (sqrt (* x 9.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.016) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = sqrt((x * 9.0)) * 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 <= 0.016d0) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    else
        tmp = sqrt((x * 9.0d0)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.016) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = Math.sqrt((x * 9.0)) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.016:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	else:
		tmp = math.sqrt((x * 9.0)) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.016)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.016)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	else
		tmp = sqrt((x * 9.0)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.016], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.016

   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%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 76.3%

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

      1. sub-neg 76.3%

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

      2. associate-*r/ 76.3%

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

      3. metadata-eval 76.3%

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

      4. metadata-eval 76.3%

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

   7. Simplified 76.3%

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

      1. add-sqr-sqrt 76.0%

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

      2. sqrt-unprod 76.3%

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

      3. associate-*r* 76.3%

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

      4. associate-*r* 76.3%

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

      5. swap-sqr 34.9%

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

      6. swap-sqr 35.0%

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

      7. metadata-eval 35.0%

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

      8. add-sqr-sqrt 35.0%

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

      9. *-commutative 35.0%

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

      10. pow2 35.0%

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

   9. Applied egg-rr 35.0%

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

       1. associate-*l* 35.0%

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

   11. Simplified 35.0%

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

   12. Taylor expanded in x around 0 76.0%

       \[\leadsto \sqrt{\color{blue}{0.1111111111111111 \cdot \frac{1}{x} - 2}} \]
   13. Step-by-step derivation

       1. sub-neg 76.0%

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

       2. associate-*r/ 75.9%

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

       3. metadata-eval 75.9%

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

       4. metadata-eval 75.9%

          \[\leadsto \sqrt{\frac{0.1111111111111111}{x} + \color{blue}{-2}} \]

   14. Simplified 75.9%

       \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x} + -2}} \]

   if 0.016 < 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* 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 48.3%

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

      1. expm1-log1p-u 22.8%

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

      2. expm1-udef 22.7%

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

      3. *-commutative 22.7%

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

      4. *-commutative 22.7%

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

      5. associate-*l* 22.7%

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

      6. metadata-eval 22.7%

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

      7. sqrt-prod 22.7%

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

   7. Applied egg-rr 22.7%

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

      1. expm1-def 22.8%

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

      2. expm1-log1p 48.3%

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

   9. Simplified 48.3%

      \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.016:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \]
5. Add Preprocessing

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%

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

   1. associate-*l* 99.3%

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

   2. associate--l+ 99.3%

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

   3. sub-neg 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 63.9%

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

   1. sub-neg 63.9%

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

   2. associate-*r/ 63.9%

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

   3. metadata-eval 63.9%

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

   4. metadata-eval 63.9%

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

7. Simplified 63.9%

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

   1. add-sqr-sqrt 37.7%

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

   2. sqrt-unprod 38.9%

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

   3. associate-*r* 38.9%

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

   4. associate-*r* 38.9%

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

   5. swap-sqr 18.4%

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

   6. swap-sqr 18.4%

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

   7. metadata-eval 18.4%

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

   8. add-sqr-sqrt 18.4%

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

   9. *-commutative 18.4%

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

   10. pow2 18.4%

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

9. Applied egg-rr 18.4%

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

    1. associate-*l* 18.4%

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

11. Simplified 18.4%

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

12. Taylor expanded in x around inf 3.3%

    \[\leadsto \sqrt{\color{blue}{9 \cdot x}} \]
13. Step-by-step derivation

    1. *-commutative 3.3%

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

14. Simplified 3.3%

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

15. Final simplification 3.3%

    \[\leadsto \sqrt{x \cdot 9} \]
16. Add Preprocessing

Alternative 12: 38.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.1111111111111111}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (/ 0.1111111111111111 x)))

C

double code(double x, double y) {
	return sqrt((0.1111111111111111 / x));
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.sqrt((0.1111111111111111 / x))

Julia

function code(x, y)
	return sqrt(Float64(0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((0.1111111111111111 / x));
end

Wolfram

code[x_, y_] := N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.3%

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

   2. associate--l+ 99.3%

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

   3. sub-neg 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. add-sqr-sqrt 55.2%

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

   2. sqrt-unprod 48.2%

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

   3. swap-sqr 25.9%

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

   4. add-sqr-sqrt 26.0%

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

   5. pow2 26.0%

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

   6. associate-+r+ 26.0%

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

   7. +-commutative 26.0%

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

   8. associate-+l+ 26.0%

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

6. Applied egg-rr 26.0%

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

   1. *-commutative 26.0%

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

   2. associate-+r+ 26.0%

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

   3. +-commutative 26.0%

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

   4. associate-+r+ 26.0%

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

8. Simplified 26.0%

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

9. Taylor expanded in x around 0 38.1%

   \[\leadsto 3 \cdot \sqrt{\color{blue}{\frac{0.012345679012345678}{x}}} \]
10. Step-by-step derivation

    1. sqrt-div 38.1%

       \[\leadsto 3 \cdot \color{blue}{\frac{\sqrt{0.012345679012345678}}{\sqrt{x}}} \]

    2. metadata-eval 38.1%

       \[\leadsto 3 \cdot \frac{\color{blue}{0.1111111111111111}}{\sqrt{x}} \]

    3. associate-*r/ 38.1%

       \[\leadsto \color{blue}{\frac{3 \cdot 0.1111111111111111}{\sqrt{x}}} \]

    4. metadata-eval 38.1%

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

    5. metadata-eval 38.1%

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

    6. sqrt-div 38.2%

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

    7. pow1/2 38.2%

       \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

11. Applied egg-rr 38.2%

    \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
12. Step-by-step derivation

    1. unpow1/2 38.2%

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

13. Simplified 38.2%

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

14. Final simplification 38.2%

    \[\leadsto \sqrt{\frac{0.1111111111111111}{x}} \]
15. Add Preprocessing

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 2024010 
(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)))