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

Percentage Accurate: 99.4% → 99.4%

Time: 11.0s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

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(y + Float64(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[(y + N[(N[(0.1111111111111111 / x), $MachinePrecision] + -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. sub-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. sqrt-prod 99.6%

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

   4. pow1/2 99.6%

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

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]
6. Step-by-step derivation

   1. unpow1/2 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9500 \lor \neg \left(y \leq 1200000000\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9500.0) (not (<= y 1200000000.0)))
   (* (sqrt (* x 9.0)) y)
   (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9500.0) || !(y <= 1200000000.0)) {
		tmp = sqrt((x * 9.0)) * y;
	} else {
		tmp = sqrt(x) * ((0.3333333333333333 / 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 <= (-9500.0d0)) .or. (.not. (y <= 1200000000.0d0))) then
        tmp = sqrt((x * 9.0d0)) * y
    else
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9500.0) || !(y <= 1200000000.0)) {
		tmp = Math.sqrt((x * 9.0)) * y;
	} else {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9500.0) or not (y <= 1200000000.0):
		tmp = math.sqrt((x * 9.0)) * y
	else:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9500.0) || !(y <= 1200000000.0))
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	else
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9500.0) || ~((y <= 1200000000.0)))
		tmp = sqrt((x * 9.0)) * y;
	else
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9500.0], N[Not[LessEqual[y, 1200000000.0]], $MachinePrecision]], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9500 or 1.2e9 < 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. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 81.0%

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

   if -9500 < y < 1.2e9

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. sub-neg 99.4%

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

      8. distribute-lft-in 99.5%

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

      9. fma-def 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.5%

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

      14. associate-*r/ 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 96.5%

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

      1. sub-neg 96.5%

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

      2. associate-*r/ 96.5%

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

      3. metadata-eval 96.5%

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

      4. metadata-eval 96.5%

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

   6. Simplified 96.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9500 \lor \neg \left(y \leq 1200000000\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 3: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \lor \neg \left(y \leq 2.25 \cdot 10^{-12}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.0) (not (<= y 2.25e-12)))
   (* (sqrt x) (- (* y 3.0) 3.0))
   (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.0) || !(y <= 2.25e-12)) {
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	} else {
		tmp = sqrt(x) * ((0.3333333333333333 / 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 <= (-2.0d0)) .or. (.not. (y <= 2.25d-12))) then
        tmp = sqrt(x) * ((y * 3.0d0) - 3.0d0)
    else
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.0) || !(y <= 2.25e-12)) {
		tmp = Math.sqrt(x) * ((y * 3.0) - 3.0);
	} else {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.0) or not (y <= 2.25e-12):
		tmp = math.sqrt(x) * ((y * 3.0) - 3.0)
	else:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.0) || !(y <= 2.25e-12))
		tmp = Float64(sqrt(x) * Float64(Float64(y * 3.0) - 3.0));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.0) || ~((y <= 2.25e-12)))
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	else
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.0], N[Not[LessEqual[y, 2.25e-12]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y * 3.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2 or 2.2499999999999999e-12 < 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. *-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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-lft-in 99.4%

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

      9. fma-def 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. associate-*r/ 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

   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 81.0%

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

   if -2 < y < 2.2499999999999999e-12

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. sub-neg 99.4%

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

      8. distribute-lft-in 99.5%

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

      9. fma-def 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.4%

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

      14. associate-*r/ 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 98.4%

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

      1. sub-neg 98.4%

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

      2. associate-*r/ 98.4%

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

      3. metadata-eval 98.4%

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

      4. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \lor \neg \left(y \leq 2.25 \cdot 10^{-12}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 4: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -210 \lor \neg \left(y \leq 2.25 \cdot 10^{-12}\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -210.0) (not (<= y 2.25e-12)))
   (* (sqrt (* x 9.0)) (+ y -1.0))
   (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -210.0) || !(y <= 2.25e-12)) {
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	} else {
		tmp = sqrt(x) * ((0.3333333333333333 / 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 <= (-210.0d0)) .or. (.not. (y <= 2.25d-12))) then
        tmp = sqrt((x * 9.0d0)) * (y + (-1.0d0))
    else
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -210.0) || !(y <= 2.25e-12)) {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	} else {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -210.0) or not (y <= 2.25e-12):
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	else:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -210.0) || !(y <= 2.25e-12))
		tmp = Float64(sqrt(Float64(x * 9.0)) * Float64(y + -1.0));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -210.0) || ~((y <= 2.25e-12)))
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	else
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -210.0], N[Not[LessEqual[y, 2.25e-12]], $MachinePrecision]], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -210 or 2.2499999999999999e-12 < 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. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 81.2%

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

   if -210 < y < 2.2499999999999999e-12

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. sub-neg 99.4%

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

      8. distribute-lft-in 99.5%

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

      9. fma-def 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.4%

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

      14. associate-*r/ 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 98.4%

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

      1. sub-neg 98.4%

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

      2. associate-*r/ 98.4%

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

      3. metadata-eval 98.4%

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

      4. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -210 \lor \neg \left(y \leq 2.25 \cdot 10^{-12}\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.4%

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

   2. associate--l+ 99.4%

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

   3. sub-neg 99.4%

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

   4. *-commutative 99.4%

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

   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. Final simplification 99.4%

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

Alternative 6: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9500 \lor \neg \left(y \leq 1550000000\right):\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9500.0) (not (<= y 1550000000.0)))
   (* 3.0 (* y (sqrt x)))
   (sqrt (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9500.0) || !(y <= 1550000000.0)) {
		tmp = 3.0 * (y * sqrt(x));
	} else {
		tmp = sqrt((0.1111111111111111 / 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 <= (-9500.0d0)) .or. (.not. (y <= 1550000000.0d0))) then
        tmp = 3.0d0 * (y * sqrt(x))
    else
        tmp = sqrt((0.1111111111111111d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9500.0) || !(y <= 1550000000.0)) {
		tmp = 3.0 * (y * Math.sqrt(x));
	} else {
		tmp = Math.sqrt((0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9500.0) or not (y <= 1550000000.0):
		tmp = 3.0 * (y * math.sqrt(x))
	else:
		tmp = math.sqrt((0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9500.0) || !(y <= 1550000000.0))
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	else
		tmp = sqrt(Float64(0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9500.0) || ~((y <= 1550000000.0)))
		tmp = 3.0 * (y * sqrt(x));
	else
		tmp = sqrt((0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9500.0], N[Not[LessEqual[y, 1550000000.0]], $MachinePrecision]], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9500 or 1.55e9 < 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. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 80.8%

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

   if -9500 < y < 1.55e9

   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* 99.3%

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

      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. Taylor expanded in x around 0 49.7%

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

      1. associate-*r* 49.9%

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

      2. expm1-log1p-u 46.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      3. expm1-udef 47.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]

      4. associate-*r/ 47.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.1111111111111111}{x}}\right)} - 1 \]

      5. *-commutative 47.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.1111111111111111}{x}\right)} - 1 \]

      6. associate-*l* 47.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.1111111111111111\right)}}{x}\right)} - 1 \]

      7. metadata-eval 47.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot \color{blue}{0.3333333333333333}}{x}\right)} - 1 \]

   6. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 46.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)\right)} \]

      2. expm1-log1p 49.7%

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

      3. rem-square-sqrt 49.7%

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

      4. associate-/l/ 49.8%

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

      5. *-commutative 49.8%

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

      6. associate-/l* 49.9%

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

      7. *-inverses 49.9%

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

      8. metadata-eval 49.9%

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

   8. Simplified 49.9%

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

      1. metadata-eval 49.9%

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

      2. sqrt-div 50.1%

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

   10. Applied egg-rr 50.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9500 \lor \neg \left(y \leq 1550000000\right):\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \]

Alternative 7: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9500 \lor \neg \left(y \leq 1650000000000\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9500.0) (not (<= y 1650000000000.0)))
   (* (sqrt (* x 9.0)) y)
   (sqrt (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9500.0) || !(y <= 1650000000000.0)) {
		tmp = sqrt((x * 9.0)) * y;
	} else {
		tmp = sqrt((0.1111111111111111 / 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 <= (-9500.0d0)) .or. (.not. (y <= 1650000000000.0d0))) then
        tmp = sqrt((x * 9.0d0)) * y
    else
        tmp = sqrt((0.1111111111111111d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9500.0) || !(y <= 1650000000000.0)) {
		tmp = Math.sqrt((x * 9.0)) * y;
	} else {
		tmp = Math.sqrt((0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9500.0) or not (y <= 1650000000000.0):
		tmp = math.sqrt((x * 9.0)) * y
	else:
		tmp = math.sqrt((0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9500.0) || !(y <= 1650000000000.0))
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	else
		tmp = sqrt(Float64(0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9500.0) || ~((y <= 1650000000000.0)))
		tmp = sqrt((x * 9.0)) * y;
	else
		tmp = sqrt((0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9500.0], N[Not[LessEqual[y, 1650000000000.0]], $MachinePrecision]], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9500 or 1.65e12 < 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. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 81.0%

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

   if -9500 < y < 1.65e12

   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* 99.3%

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

      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. Taylor expanded in x around 0 49.7%

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

      1. associate-*r* 49.9%

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

      2. expm1-log1p-u 46.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      3. expm1-udef 47.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]

      4. associate-*r/ 47.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.1111111111111111}{x}}\right)} - 1 \]

      5. *-commutative 47.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.1111111111111111}{x}\right)} - 1 \]

      6. associate-*l* 47.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.1111111111111111\right)}}{x}\right)} - 1 \]

      7. metadata-eval 47.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot \color{blue}{0.3333333333333333}}{x}\right)} - 1 \]

   6. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 46.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)\right)} \]

      2. expm1-log1p 49.7%

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

      3. rem-square-sqrt 49.7%

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

      4. associate-/l/ 49.8%

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

      5. *-commutative 49.8%

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

      6. associate-/l* 49.9%

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

      7. *-inverses 49.9%

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

      8. metadata-eval 49.9%

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

   8. Simplified 49.9%

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

      1. metadata-eval 49.9%

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

      2. sqrt-div 50.1%

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

   10. Applied egg-rr 50.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9500 \lor \neg \left(y \leq 1650000000000\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \end{array} \]

Alternative 8: 38.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(x \cdot 9\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (pow (* x 9.0) -0.5))

C

double code(double x, double y) {
	return pow((x * 9.0), -0.5);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * 9.0d0) ** (-0.5d0)
end function

Java

public static double code(double x, double y) {
	return Math.pow((x * 9.0), -0.5);
}

Python

def code(x, y):
	return math.pow((x * 9.0), -0.5)

Julia

function code(x, y)
	return Float64(x * 9.0) ^ -0.5
end

MATLAB

function tmp = code(x, y)
	tmp = (x * 9.0) ^ -0.5;
end

Wolfram

code[x_, y_] := N[Power[N[(x * 9.0), $MachinePrecision], -0.5], $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%

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

   2. associate--l+ 99.4%

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

   3. sub-neg 99.4%

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

   4. *-commutative 99.4%

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

   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. Taylor expanded in x around 0 33.8%

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

   1. associate-*r* 33.9%

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

   2. expm1-log1p-u 31.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

   3. expm1-udef 31.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]

   4. associate-*r/ 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.1111111111111111}{x}}\right)} - 1 \]

   5. *-commutative 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.1111111111111111}{x}\right)} - 1 \]

   6. associate-*l* 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.1111111111111111\right)}}{x}\right)} - 1 \]

   7. metadata-eval 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot \color{blue}{0.3333333333333333}}{x}\right)} - 1 \]

6. Applied egg-rr 31.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 31.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)\right)} \]

   2. expm1-log1p 33.8%

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

   3. rem-square-sqrt 33.8%

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

   4. associate-/l/ 33.9%

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

   5. *-commutative 33.9%

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

   6. associate-/l* 33.9%

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

   7. *-inverses 33.9%

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

   8. metadata-eval 33.9%

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

8. Simplified 33.9%

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

   1. clear-num 33.9%

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

   2. div-inv 33.9%

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

   3. metadata-eval 33.9%

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

   4. metadata-eval 33.9%

      \[\leadsto \frac{1}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \]

   5. sqrt-prod 33.9%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x \cdot 9}}} \]

   6. pow1/2 33.9%

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

   7. metadata-eval 33.9%

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

   8. pow-div 33.9%

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

   9. pow1 33.9%

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

   10. pow1/2 33.9%

       \[\leadsto \frac{1}{\frac{x \cdot 9}{\color{blue}{\sqrt{x \cdot 9}}}} \]

   11. sqrt-prod 33.8%

       \[\leadsto \frac{1}{\frac{x \cdot 9}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}}} \]

   12. metadata-eval 33.8%

       \[\leadsto \frac{1}{\frac{x \cdot 9}{\sqrt{x} \cdot \color{blue}{3}}} \]

   13. *-commutative 33.8%

       \[\leadsto \frac{1}{\frac{x \cdot 9}{\color{blue}{3 \cdot \sqrt{x}}}} \]

   14. clear-num 33.9%

       \[\leadsto \color{blue}{\frac{3 \cdot \sqrt{x}}{x \cdot 9}} \]

   15. *-commutative 33.9%

       \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot 3}}{x \cdot 9} \]

   16. metadata-eval 33.9%

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

   17. sqrt-prod 34.0%

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

   18. pow1/2 34.0%

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

   19. pow1 34.0%

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

   20. pow-div 34.0%

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

   21. metadata-eval 34.0%

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

10. Applied egg-rr 34.0%

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

11. Final simplification 34.0%

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

Alternative 9: 38.1% 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* 99.4%

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

   2. associate--l+ 99.4%

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

   3. sub-neg 99.4%

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

   4. *-commutative 99.4%

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

   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. Taylor expanded in x around 0 33.8%

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

   1. associate-*r* 33.9%

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

   2. expm1-log1p-u 31.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

   3. expm1-udef 31.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]

   4. associate-*r/ 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.1111111111111111}{x}}\right)} - 1 \]

   5. *-commutative 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.1111111111111111}{x}\right)} - 1 \]

   6. associate-*l* 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.1111111111111111\right)}}{x}\right)} - 1 \]

   7. metadata-eval 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot \color{blue}{0.3333333333333333}}{x}\right)} - 1 \]

6. Applied egg-rr 31.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 31.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)\right)} \]

   2. expm1-log1p 33.8%

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

   3. rem-square-sqrt 33.8%

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

   4. associate-/l/ 33.9%

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

   5. *-commutative 33.9%

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

   6. associate-/l* 33.9%

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

   7. *-inverses 33.9%

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

   8. metadata-eval 33.9%

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

8. Simplified 33.9%

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

   1. metadata-eval 33.9%

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

   2. sqrt-div 34.0%

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

10. Applied egg-rr 34.0%

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

11. Final simplification 34.0%

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

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 2023279 
(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)))