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

Percentage Accurate: 99.7% → 99.7%

Time: 7.4s

Alternatives: 6

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

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

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]

Derivation

1. Initial program 99.7%

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

2. Final simplification 99.7%

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

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. *-un-lft-identity 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.6%

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

   4. pow1/2 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

   5. pow-flip 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

   6. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

   7. div-inv 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{-0.5} \cdot \color{blue}{\left(y \cdot \frac{1}{3}\right)} \]

   8. metadata-eval 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{-0.5} \cdot \left(y \cdot \color{blue}{0.3333333333333333}\right) \]

4. Applied egg-rr 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)} \]
5. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{\left(y \cdot 0.3333333333333333\right) \cdot {x}^{-0.5}} \]

   2. associate-*l* 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{y \cdot \left(0.3333333333333333 \cdot {x}^{-0.5}\right)} \]

6. Simplified 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \color{blue}{y \cdot \left(0.3333333333333333 \cdot {x}^{-0.5}\right)} \]
7. Step-by-step derivation

   1. metadata-eval 99.6%

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

   2. metadata-eval 99.6%

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

   3. sqrt-pow1 99.6%

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

   4. inv-pow 99.6%

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

   5. sqrt-prod 99.6%

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

   6. *-commutative 99.6%

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

   7. pow1/2 99.6%

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

   8. associate-*l/ 99.6%

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

   9. metadata-eval 99.6%

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

8. Applied egg-rr 99.6%

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

   1. unpow1/2 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * 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. 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. Final simplification 99.7%

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

Alternative 5: 62.8% 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 64.0%

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

   1. clear-num 63.9%

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

   2. associate-/r/ 63.9%

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

6. Applied egg-rr 63.9%

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

   1. associate-/r/ 63.9%

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

   2. div-inv 64.0%

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

   3. metadata-eval 64.0%

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

8. Applied egg-rr 64.0%

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

9. Final simplification 64.0%

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

Alternative 6: 62.8% 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 64.0%

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

5. Final simplification 64.0%

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

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 2023333 
(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)))))