Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D

Percentage Accurate: 99.7% → 99.7%

Time: 7.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

3. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- 1.0 (/ 0.1111111111111111 x)) (* -0.3333333333333333 (/ y (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) + ((-0.3333333333333333d0) * (y / sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) + Float64(-0.3333333333333333 * Float64(y / sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

4. Final simplification 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 99.7%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}} \cdot y \]

   5. metadata-eval 99.6%

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

   6. sqrt-div 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

3. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 65.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{1 - \frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{1 - \frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e+95)
   (/
    (- 1.0 (* (/ 0.1111111111111111 x) (/ 0.1111111111111111 x)))
    (- 1.0 (/ 0.1111111111111111 x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+95) {
		tmp = (1.0 - ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 / x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.2d+95)) then
        tmp = (1.0d0 - ((0.1111111111111111d0 / x) * (0.1111111111111111d0 / x))) / (1.0d0 - (0.1111111111111111d0 / x))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+95) {
		tmp = (1.0 - ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 / x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e+95:
		tmp = (1.0 - ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 / x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e+95)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.1111111111111111 / x) * Float64(0.1111111111111111 / x))) / Float64(1.0 - Float64(0.1111111111111111 / x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e+95)
		tmp = (1.0 - ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (1.0 - (0.1111111111111111 / x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e+95], N[(N[(1.0 - N[(N[(0.1111111111111111 / x), $MachinePrecision] * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1999999999999999e95

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. associate--l+ 99.5%

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

      4. neg-mul-1 99.5%

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

      5. sub-neg 99.5%

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

      6. distribute-frac-neg 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-/r* 99.5%

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

      9. associate-*r/ 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

      13. neg-mul-1 99.5%

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

      14. *-commutative 99.5%

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

      15. associate-*r/ 99.5%

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

      16. fma-def 99.5%

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

      17. associate-/r* 99.5%

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

      18. metadata-eval 99.5%

          \[\leadsto \frac{-0.1111111111111111}{x} + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, 1\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, 1\right)} \]

   4. Taylor expanded in y around 0 2.4%

      \[\leadsto \frac{-0.1111111111111111}{x} + \color{blue}{1} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 21.0%

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

      3. frac-times 21.0%

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

      4. metadata-eval 21.0%

         \[\leadsto \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} + 1 \]

      5. metadata-eval 21.0%

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

      6. frac-times 21.0%

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

      7. metadata-eval 21.0%

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

      8. associate-*l/ 21.0%

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

      9. metadata-eval 21.0%

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

      10. associate-*l/ 21.0%

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

      11. swap-sqr 22.9%

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

      12. inv-pow 22.9%

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

      13. inv-pow 22.9%

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

      14. pow-prod-up 22.9%

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

      15. metadata-eval 22.9%

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

      16. metadata-eval 22.9%

          \[\leadsto \sqrt{{x}^{-2} \cdot \color{blue}{0.012345679012345678}} + 1 \]

   6. Applied egg-rr 22.9%

      \[\leadsto \color{blue}{\sqrt{{x}^{-2} \cdot 0.012345679012345678}} + 1 \]
   7. Step-by-step derivation

      1. metadata-eval 22.9%

         \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot -1\right)}} \cdot 0.012345679012345678} + 1 \]

      2. pow-sqr 22.9%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \cdot 0.012345679012345678} + 1 \]

      3. unpow-1 22.9%

         \[\leadsto \sqrt{\left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \cdot 0.012345679012345678} + 1 \]

      4. unpow-1 22.9%

         \[\leadsto \sqrt{\left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \cdot 0.012345679012345678} + 1 \]

      5. metadata-eval 22.9%

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

      6. swap-sqr 21.0%

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

      7. associate-*l/ 21.0%

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

      8. metadata-eval 21.0%

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

      9. associate-*l/ 21.0%

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

      10. metadata-eval 21.0%

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

      11. rem-sqrt-square 5.8%

          \[\leadsto \color{blue}{\left|\frac{-0.1111111111111111}{x}\right|} + 1 \]

   8. Simplified 5.8%

      \[\leadsto \color{blue}{\left|\frac{-0.1111111111111111}{x}\right|} + 1 \]
   9. Step-by-step derivation

      1. +-commutative 5.8%

         \[\leadsto \color{blue}{1 + \left|\frac{-0.1111111111111111}{x}\right|} \]

      2. fabs-div 5.8%

         \[\leadsto 1 + \color{blue}{\frac{\left|-0.1111111111111111\right|}{\left|x\right|}} \]

      3. metadata-eval 5.8%

         \[\leadsto 1 + \frac{\color{blue}{0.1111111111111111}}{\left|x\right|} \]

      4. add-sqr-sqrt 5.8%

         \[\leadsto 1 + \frac{0.1111111111111111}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \]

      5. fabs-sqr 5.8%

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

      6. add-sqr-sqrt 5.8%

         \[\leadsto 1 + \frac{0.1111111111111111}{\color{blue}{x}} \]

      7. metadata-eval 5.8%

         \[\leadsto 1 + \frac{\color{blue}{0.1111111111111111 \cdot 1}}{x} \]

      8. associate-*l/ 5.8%

         \[\leadsto 1 + \color{blue}{\frac{0.1111111111111111}{x} \cdot 1} \]

      9. metadata-eval 5.8%

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

      10. distribute-neg-frac 5.8%

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

      11. cancel-sign-sub-inv 5.8%

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

      12. *-rgt-identity 5.8%

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

   10. Applied egg-rr 5.8%

       \[\leadsto \color{blue}{1 - \frac{-0.1111111111111111}{x}} \]
   11. Step-by-step derivation

       1. sub-neg 5.8%

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

       2. flip-+ 21.0%

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

       3. metadata-eval 21.0%

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

       4. distribute-neg-frac 21.0%

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

       5. metadata-eval 21.0%

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

       6. distribute-neg-frac 21.0%

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

       7. metadata-eval 21.0%

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

       8. distribute-neg-frac 21.0%

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

       9. metadata-eval 21.0%

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

   12. Applied egg-rr 21.0%

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

   if -2.1999999999999999e95 < y

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. neg-mul-1 99.7%

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

      5. sub-neg 99.7%

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

      6. distribute-frac-neg 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.7%

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

      9. associate-*r/ 99.7%

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

      10. metadata-eval 99.7%

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

      11. metadata-eval 99.7%

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

      12. +-commutative 99.7%

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

      13. neg-mul-1 99.7%

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

      14. *-commutative 99.7%

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

      15. associate-*r/ 99.7%

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

      16. fma-def 99.7%

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

      17. associate-/r* 99.6%

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

      18. metadata-eval 99.6%

          \[\leadsto \frac{-0.1111111111111111}{x} + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, 1\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, 1\right)} \]

   4. Taylor expanded in y around 0 74.9%

      \[\leadsto \frac{-0.1111111111111111}{x} + \color{blue}{1} \]
   5. Step-by-step derivation

      1. clear-num 74.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}} + 1 \]

      2. associate-/r/ 74.9%

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

   6. Applied egg-rr 74.9%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot -0.1111111111111111} + 1 \]
   7. Step-by-step derivation

      1. associate-/r/ 74.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}} + 1 \]

      2. clear-num 74.9%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} + 1 \]

      3. add-sqr-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. add-sqr-sqrt 36.8%

         \[\leadsto \left|\color{blue}{\frac{-0.1111111111111111}{x}}\right| + 1 \]

      6. fabs-div 36.8%

         \[\leadsto \color{blue}{\frac{\left|-0.1111111111111111\right|}{\left|x\right|}} + 1 \]

      7. metadata-eval 36.8%

         \[\leadsto \frac{\color{blue}{0.1111111111111111}}{\left|x\right|} + 1 \]

      8. add-sqr-sqrt 36.8%

         \[\leadsto \frac{0.1111111111111111}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} + 1 \]

      9. fabs-sqr 36.8%

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

      10. metadata-eval 36.8%

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

      11. add-sqr-sqrt 36.8%

          \[\leadsto \frac{--0.1111111111111111}{\color{blue}{x}} + 1 \]

      12. distribute-neg-frac 36.8%

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

      13. clear-num 36.8%

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

      14. distribute-neg-frac 36.8%

          \[\leadsto \color{blue}{\frac{-1}{\frac{x}{-0.1111111111111111}}} + 1 \]

      15. metadata-eval 36.8%

          \[\leadsto \frac{\color{blue}{-1}}{\frac{x}{-0.1111111111111111}} + 1 \]

      16. clear-num 36.8%

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{\frac{-0.1111111111111111}{x}}}} + 1 \]

      17. add-sqr-sqrt 0.0%

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

      18. fabs-sqr 0.0%

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

      19. add-sqr-sqrt 74.9%

          \[\leadsto \frac{-1}{\frac{1}{\left|\color{blue}{\frac{-0.1111111111111111}{x}}\right|}} + 1 \]

      20. fabs-div 74.9%

          \[\leadsto \frac{-1}{\frac{1}{\color{blue}{\frac{\left|-0.1111111111111111\right|}{\left|x\right|}}}} + 1 \]

      21. metadata-eval 74.9%

          \[\leadsto \frac{-1}{\frac{1}{\frac{\color{blue}{0.1111111111111111}}{\left|x\right|}}} + 1 \]

      22. add-sqr-sqrt 74.7%

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

      23. fabs-sqr 74.7%

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

      24. add-sqr-sqrt 74.9%

          \[\leadsto \frac{-1}{\frac{1}{\frac{0.1111111111111111}{\color{blue}{x}}}} + 1 \]

      25. clear-num 74.9%

          \[\leadsto \frac{-1}{\color{blue}{\frac{x}{0.1111111111111111}}} + 1 \]

      26. div-inv 74.9%

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

      27. metadata-eval 74.9%

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

   8. Applied egg-rr 74.9%

      \[\leadsto \color{blue}{\frac{-1}{x \cdot 9}} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{1 - \frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{1 - \frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 6: 63.0% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{x \cdot 9} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--l+ 99.7%

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

   4. neg-mul-1 99.7%

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

   5. sub-neg 99.7%

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

   6. distribute-frac-neg 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-/r* 99.7%

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

   9. associate-*r/ 99.7%

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

   10. metadata-eval 99.7%

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

   11. metadata-eval 99.7%

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

   12. +-commutative 99.7%

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

   13. neg-mul-1 99.7%

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

   14. *-commutative 99.7%

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

   15. associate-*r/ 99.6%

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

   16. fma-def 99.6%

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

   17. associate-/r* 99.6%

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

   18. metadata-eval 99.6%

       \[\leadsto \frac{-0.1111111111111111}{x} + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, 1\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, 1\right)} \]

4. Taylor expanded in y around 0 61.0%

   \[\leadsto \frac{-0.1111111111111111}{x} + \color{blue}{1} \]
5. Step-by-step derivation

   1. clear-num 61.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}} + 1 \]

   2. associate-/r/ 61.0%

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

6. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{\frac{1}{x} \cdot -0.1111111111111111} + 1 \]
7. Step-by-step derivation

   1. associate-/r/ 61.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}} + 1 \]

   2. clear-num 61.0%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} + 1 \]

   3. add-sqr-sqrt 0.0%

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

   4. fabs-sqr 0.0%

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

   5. add-sqr-sqrt 30.9%

      \[\leadsto \left|\color{blue}{\frac{-0.1111111111111111}{x}}\right| + 1 \]

   6. fabs-div 30.9%

      \[\leadsto \color{blue}{\frac{\left|-0.1111111111111111\right|}{\left|x\right|}} + 1 \]

   7. metadata-eval 30.9%

      \[\leadsto \frac{\color{blue}{0.1111111111111111}}{\left|x\right|} + 1 \]

   8. add-sqr-sqrt 30.9%

      \[\leadsto \frac{0.1111111111111111}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} + 1 \]

   9. fabs-sqr 30.9%

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

   10. metadata-eval 30.9%

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

   11. add-sqr-sqrt 30.9%

       \[\leadsto \frac{--0.1111111111111111}{\color{blue}{x}} + 1 \]

   12. distribute-neg-frac 30.9%

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

   13. clear-num 30.9%

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

   14. distribute-neg-frac 30.9%

       \[\leadsto \color{blue}{\frac{-1}{\frac{x}{-0.1111111111111111}}} + 1 \]

   15. metadata-eval 30.9%

       \[\leadsto \frac{\color{blue}{-1}}{\frac{x}{-0.1111111111111111}} + 1 \]

   16. clear-num 30.9%

       \[\leadsto \frac{-1}{\color{blue}{\frac{1}{\frac{-0.1111111111111111}{x}}}} + 1 \]

   17. add-sqr-sqrt 0.0%

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

   18. fabs-sqr 0.0%

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

   19. add-sqr-sqrt 61.0%

       \[\leadsto \frac{-1}{\frac{1}{\left|\color{blue}{\frac{-0.1111111111111111}{x}}\right|}} + 1 \]

   20. fabs-div 61.0%

       \[\leadsto \frac{-1}{\frac{1}{\color{blue}{\frac{\left|-0.1111111111111111\right|}{\left|x\right|}}}} + 1 \]

   21. metadata-eval 61.0%

       \[\leadsto \frac{-1}{\frac{1}{\frac{\color{blue}{0.1111111111111111}}{\left|x\right|}}} + 1 \]

   22. add-sqr-sqrt 60.9%

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

   23. fabs-sqr 60.9%

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

   24. add-sqr-sqrt 61.0%

       \[\leadsto \frac{-1}{\frac{1}{\frac{0.1111111111111111}{\color{blue}{x}}}} + 1 \]

   25. clear-num 61.0%

       \[\leadsto \frac{-1}{\color{blue}{\frac{x}{0.1111111111111111}}} + 1 \]

   26. div-inv 61.1%

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

   27. metadata-eval 61.1%

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

8. Applied egg-rr 61.1%

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

9. Final simplification 61.1%

   \[\leadsto 1 + \frac{-1}{x \cdot 9} \]

Alternative 7: 63.0% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--l+ 99.7%

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

   4. neg-mul-1 99.7%

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

   5. sub-neg 99.7%

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

   6. distribute-frac-neg 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-/r* 99.7%

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

   9. associate-*r/ 99.7%

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

   10. metadata-eval 99.7%

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

   11. metadata-eval 99.7%

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

   12. +-commutative 99.7%

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

   13. neg-mul-1 99.7%

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

   14. *-commutative 99.7%

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

   15. associate-*r/ 99.6%

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

   16. fma-def 99.6%

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

   17. associate-/r* 99.6%

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

   18. metadata-eval 99.6%

       \[\leadsto \frac{-0.1111111111111111}{x} + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, 1\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, 1\right)} \]

4. Taylor expanded in y around 0 61.0%

   \[\leadsto \frac{-0.1111111111111111}{x} + \color{blue}{1} \]

5. Final simplification 61.0%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]

Alternative 8: 31.9% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--l+ 99.7%

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

   4. neg-mul-1 99.7%

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

   5. sub-neg 99.7%

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

   6. distribute-frac-neg 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-/r* 99.7%

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

   9. associate-*r/ 99.7%

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

   10. metadata-eval 99.7%

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

   11. metadata-eval 99.7%

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

   12. +-commutative 99.7%

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

   13. neg-mul-1 99.7%

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

   14. *-commutative 99.7%

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

   15. associate-*r/ 99.6%

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

   16. fma-def 99.6%

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

   17. associate-/r* 99.6%

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

   18. metadata-eval 99.6%

       \[\leadsto \frac{-0.1111111111111111}{x} + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, 1\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, 1\right)} \]

4. Taylor expanded in y around 0 61.0%

   \[\leadsto \frac{-0.1111111111111111}{x} + \color{blue}{1} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 33.8%

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

   3. frac-times 33.8%

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

   4. metadata-eval 33.8%

      \[\leadsto \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} + 1 \]

   5. metadata-eval 33.8%

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

   6. frac-times 33.8%

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

   7. metadata-eval 33.8%

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

   8. associate-*l/ 33.8%

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

   9. metadata-eval 33.8%

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

   10. associate-*l/ 33.8%

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

   11. swap-sqr 34.1%

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

   12. inv-pow 34.1%

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

   13. inv-pow 34.1%

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

   14. pow-prod-up 34.1%

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

   15. metadata-eval 34.1%

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

   16. metadata-eval 34.1%

       \[\leadsto \sqrt{{x}^{-2} \cdot \color{blue}{0.012345679012345678}} + 1 \]

6. Applied egg-rr 34.1%

   \[\leadsto \color{blue}{\sqrt{{x}^{-2} \cdot 0.012345679012345678}} + 1 \]
7. Step-by-step derivation

   1. metadata-eval 34.1%

      \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot -1\right)}} \cdot 0.012345679012345678} + 1 \]

   2. pow-sqr 34.1%

      \[\leadsto \sqrt{\color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \cdot 0.012345679012345678} + 1 \]

   3. unpow-1 34.1%

      \[\leadsto \sqrt{\left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \cdot 0.012345679012345678} + 1 \]

   4. unpow-1 34.1%

      \[\leadsto \sqrt{\left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \cdot 0.012345679012345678} + 1 \]

   5. metadata-eval 34.1%

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

   6. swap-sqr 33.8%

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

   7. associate-*l/ 33.8%

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

   8. metadata-eval 33.8%

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

   9. associate-*l/ 33.8%

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

   10. metadata-eval 33.8%

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

   11. rem-sqrt-square 30.9%

       \[\leadsto \color{blue}{\left|\frac{-0.1111111111111111}{x}\right|} + 1 \]

8. Simplified 30.9%

   \[\leadsto \color{blue}{\left|\frac{-0.1111111111111111}{x}\right|} + 1 \]
9. Step-by-step derivation

   1. +-commutative 30.9%

      \[\leadsto \color{blue}{1 + \left|\frac{-0.1111111111111111}{x}\right|} \]

   2. fabs-div 30.9%

      \[\leadsto 1 + \color{blue}{\frac{\left|-0.1111111111111111\right|}{\left|x\right|}} \]

   3. metadata-eval 30.9%

      \[\leadsto 1 + \frac{\color{blue}{0.1111111111111111}}{\left|x\right|} \]

   4. add-sqr-sqrt 30.9%

      \[\leadsto 1 + \frac{0.1111111111111111}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \]

   5. fabs-sqr 30.9%

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

   6. add-sqr-sqrt 30.9%

      \[\leadsto 1 + \frac{0.1111111111111111}{\color{blue}{x}} \]

   7. metadata-eval 30.9%

      \[\leadsto 1 + \frac{\color{blue}{0.1111111111111111 \cdot 1}}{x} \]

   8. associate-*l/ 30.9%

      \[\leadsto 1 + \color{blue}{\frac{0.1111111111111111}{x} \cdot 1} \]

   9. metadata-eval 30.9%

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

   10. distribute-neg-frac 30.9%

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

   11. cancel-sign-sub-inv 30.9%

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

   12. *-rgt-identity 30.9%

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

10. Applied egg-rr 30.9%

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

11. Taylor expanded in x around inf 30.8%

    \[\leadsto \color{blue}{1} \]

12. Final simplification 30.8%

    \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023334 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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