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

Percentage Accurate: 99.7% → 99.7%

Time: 8.7s

Alternatives: 12

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

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

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(1.0 + Float64(Float64(-1.0 / Float64(x * 9.0)) - Float64(Float64(y / 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[(1.0 + N[(N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-/r* 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \left(\frac{-1}{x \cdot 9} - \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ 1.0 (- (/ -1.0 (* x 9.0)) (/ (* y 0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	return 1.0 + ((-1.0 / (x * 9.0)) - ((y * 0.3333333333333333) / 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 * 0.3333333333333333d0) / sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((-1.0 / (x * 9.0)) - ((y * 0.3333333333333333) / Math.sqrt(x)));
}

Python

def code(x, y):
	return 1.0 + ((-1.0 / (x * 9.0)) - ((y * 0.3333333333333333) / math.sqrt(x)))

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(-1.0 / Float64(x * 9.0)) - Float64(Float64(y * 0.3333333333333333) / sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((-1.0 / (x * 9.0)) - ((y * 0.3333333333333333) / sqrt(x)));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - N[(N[(y * 0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-/r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}}\right) \]

5. Applied egg-rr 99.7%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333 \cdot \frac{y}{\sqrt{x}}}\right) \]
6. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{0.3333333333333333 \cdot y}{\sqrt{x}}}\right) \]

   2. *-commutative 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \frac{\color{blue}{y \cdot 0.3333333333333333}}{\sqrt{x}}\right) \]

7. Simplified 99.7%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{y \cdot 0.3333333333333333}{\sqrt{x}}}\right) \]

8. Final simplification 99.7%

   \[\leadsto 1 + \left(\frac{-1}{x \cdot 9} - \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\right) \]

Alternative 3: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.36 \cdot 10^{+50}:\\ \;\;\;\;1 - \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y \cdot 0.3333333333333333\right) \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.36e+50)
   (- 1.0 (/ (* y 0.3333333333333333) (sqrt x)))
   (if (<= y 6.2e+42)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (* (* y 0.3333333333333333) (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.36e+50) {
		tmp = 1.0 - ((y * 0.3333333333333333) / sqrt(x));
	} else if (y <= 6.2e+42) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y * 0.3333333333333333) * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.36d+50)) then
        tmp = 1.0d0 - ((y * 0.3333333333333333d0) / sqrt(x))
    else if (y <= 6.2d+42) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - ((y * 0.3333333333333333d0) * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.36e+50) {
		tmp = 1.0 - ((y * 0.3333333333333333) / Math.sqrt(x));
	} else if (y <= 6.2e+42) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((y * 0.3333333333333333) * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.36e+50:
		tmp = 1.0 - ((y * 0.3333333333333333) / math.sqrt(x))
	elif y <= 6.2e+42:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - ((y * 0.3333333333333333) * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.36e+50)
		tmp = Float64(1.0 - Float64(Float64(y * 0.3333333333333333) / sqrt(x)));
	elseif (y <= 6.2e+42)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(y * 0.3333333333333333) * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.36e+50)
		tmp = 1.0 - ((y * 0.3333333333333333) / sqrt(x));
	elseif (y <= 6.2e+42)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - ((y * 0.3333333333333333) * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.36e+50], N[(1.0 - N[(N[(y * 0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+42], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y * 0.3333333333333333), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.36e50

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. associate-/r* 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. times-frac 99.4%

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

      4. metadata-eval 99.4%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}}\right) \]

   5. Applied egg-rr 99.4%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333 \cdot \frac{y}{\sqrt{x}}}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{0.3333333333333333 \cdot y}{\sqrt{x}}}\right) \]

      2. *-commutative 99.5%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \frac{\color{blue}{y \cdot 0.3333333333333333}}{\sqrt{x}}\right) \]

   7. Simplified 99.5%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{y \cdot 0.3333333333333333}{\sqrt{x}}}\right) \]

   8. Taylor expanded in x around 0 99.5%

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

   9. Taylor expanded in y around inf 94.8%

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

       1. associate-*r* 94.7%

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

   11. Simplified 94.7%

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

       1. metadata-eval 94.7%

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

       2. associate-/r/ 94.7%

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

       3. sqrt-div 94.8%

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

       4. metadata-eval 94.8%

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

       5. pow1/2 94.8%

          \[\leadsto 1 - \frac{1}{\frac{3}{y}} \cdot \frac{1}{\color{blue}{{x}^{0.5}}} \]

       6. div-inv 95.0%

          \[\leadsto 1 - \color{blue}{\frac{\frac{1}{\frac{3}{y}}}{{x}^{0.5}}} \]

       7. pow1/2 95.0%

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

       8. associate-/r/ 94.9%

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

       9. metadata-eval 94.9%

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

       10. *-commutative 94.9%

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

   13. Applied egg-rr 94.9%

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

   if -1.36e50 < y < 6.2000000000000003e42

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r* 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 96.2%

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

      1. associate-*r/ 96.2%

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

      2. metadata-eval 96.2%

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

   6. Simplified 96.2%

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

      1. div-inv 96.2%

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

      2. metadata-eval 96.2%

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

      3. inv-pow 96.2%

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

      4. unpow-prod-down 96.4%

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

      5. *-commutative 96.4%

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

      6. unpow-prod-down 96.2%

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

      7. inv-pow 96.2%

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

      8. metadata-eval 96.2%

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

   8. Applied egg-rr 96.2%

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

      1. associate-*l/ 96.2%

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

      2. metadata-eval 96.2%

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

      3. clear-num 96.3%

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

      4. div-inv 96.4%

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

      5. metadata-eval 96.4%

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

   10. Applied egg-rr 96.4%

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

   if 6.2000000000000003e42 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-/r* 99.4%

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

   3. Simplified 99.4%

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

      1. associate-/r* 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. times-frac 99.4%

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

      4. metadata-eval 99.4%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}}\right) \]

   5. Applied egg-rr 99.4%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333 \cdot \frac{y}{\sqrt{x}}}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{0.3333333333333333 \cdot y}{\sqrt{x}}}\right) \]

      2. *-commutative 99.4%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \frac{\color{blue}{y \cdot 0.3333333333333333}}{\sqrt{x}}\right) \]

   7. Simplified 99.4%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{y \cdot 0.3333333333333333}{\sqrt{x}}}\right) \]

   8. Taylor expanded in x around 0 99.4%

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

   9. Taylor expanded in y around inf 92.0%

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

       1. associate-*r* 92.0%

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

   11. Simplified 92.0%

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

       1. expm1-log1p-u 89.9%

          \[\leadsto 1 - \left(0.3333333333333333 \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)\right)} \]

       2. expm1-udef 54.2%

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

       3. inv-pow 54.2%

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

       4. sqrt-pow1 54.2%

          \[\leadsto 1 - \left(0.3333333333333333 \cdot y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

       5. metadata-eval 54.2%

          \[\leadsto 1 - \left(0.3333333333333333 \cdot y\right) \cdot \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   13. Applied egg-rr 54.2%

       \[\leadsto 1 - \left(0.3333333333333333 \cdot y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 89.9%

          \[\leadsto 1 - \left(0.3333333333333333 \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 92.1%

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

   15. Simplified 92.1%

       \[\leadsto 1 - \left(0.3333333333333333 \cdot y\right) \cdot \color{blue}{{x}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.36 \cdot 10^{+50}:\\ \;\;\;\;1 - \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y \cdot 0.3333333333333333\right) \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 4: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+49} \lor \neg \left(y \leq 4.9 \cdot 10^{+42}\right):\\ \;\;\;\;1 - \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.3e+49) (not (<= y 4.9e+42)))
   (- 1.0 (/ (* y 0.3333333333333333) (sqrt x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3e+49) || !(y <= 4.9e+42)) {
		tmp = 1.0 - ((y * 0.3333333333333333) / sqrt(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 <= (-1.3d+49)) .or. (.not. (y <= 4.9d+42))) then
        tmp = 1.0d0 - ((y * 0.3333333333333333d0) / sqrt(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 <= -1.3e+49) || !(y <= 4.9e+42)) {
		tmp = 1.0 - ((y * 0.3333333333333333) / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.3e+49) or not (y <= 4.9e+42):
		tmp = 1.0 - ((y * 0.3333333333333333) / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.3e+49) || !(y <= 4.9e+42))
		tmp = Float64(1.0 - Float64(Float64(y * 0.3333333333333333) / sqrt(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 <= -1.3e+49) || ~((y <= 4.9e+42)))
		tmp = 1.0 - ((y * 0.3333333333333333) / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.3e+49], N[Not[LessEqual[y, 4.9e+42]], $MachinePrecision]], N[(1.0 - N[(N[(y * 0.3333333333333333), $MachinePrecision] / N[Sqrt[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 < -1.29999999999999994e49 or 4.9000000000000002e42 < y

   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. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r* 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. times-frac 99.4%

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

      4. metadata-eval 99.4%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}}\right) \]

   5. Applied egg-rr 99.4%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333 \cdot \frac{y}{\sqrt{x}}}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{0.3333333333333333 \cdot y}{\sqrt{x}}}\right) \]

      2. *-commutative 99.5%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \frac{\color{blue}{y \cdot 0.3333333333333333}}{\sqrt{x}}\right) \]

   7. Simplified 99.5%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{y \cdot 0.3333333333333333}{\sqrt{x}}}\right) \]

   8. Taylor expanded in x around 0 99.4%

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

   9. Taylor expanded in y around inf 93.4%

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

       1. associate-*r* 93.4%

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

   11. Simplified 93.4%

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

       1. metadata-eval 93.4%

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

       2. associate-/r/ 93.4%

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

       3. sqrt-div 93.3%

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

       4. metadata-eval 93.3%

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

       5. pow1/2 93.3%

          \[\leadsto 1 - \frac{1}{\frac{3}{y}} \cdot \frac{1}{\color{blue}{{x}^{0.5}}} \]

       6. div-inv 93.4%

          \[\leadsto 1 - \color{blue}{\frac{\frac{1}{\frac{3}{y}}}{{x}^{0.5}}} \]

       7. pow1/2 93.4%

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

       8. associate-/r/ 93.4%

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

       9. metadata-eval 93.4%

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

       10. *-commutative 93.4%

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

   13. Applied egg-rr 93.4%

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

   if -1.29999999999999994e49 < y < 4.9000000000000002e42

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r* 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 96.2%

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

      1. associate-*r/ 96.2%

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

      2. metadata-eval 96.2%

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

   6. Simplified 96.2%

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

      1. div-inv 96.2%

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

      2. metadata-eval 96.2%

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

      3. inv-pow 96.2%

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

      4. unpow-prod-down 96.4%

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

      5. *-commutative 96.4%

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

      6. unpow-prod-down 96.2%

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

      7. inv-pow 96.2%

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

      8. metadata-eval 96.2%

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

   8. Applied egg-rr 96.2%

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

      1. associate-*l/ 96.2%

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

      2. metadata-eval 96.2%

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

      3. clear-num 96.3%

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

      4. div-inv 96.4%

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

      5. metadata-eval 96.4%

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

   10. Applied egg-rr 96.4%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+49} \lor \neg \left(y \leq 4.9 \cdot 10^{+42}\right):\\ \;\;\;\;1 - \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-/r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}}\right) \]

5. Applied egg-rr 99.7%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333 \cdot \frac{y}{\sqrt{x}}}\right) \]
6. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{0.3333333333333333 \cdot y}{\sqrt{x}}}\right) \]

   2. *-commutative 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \frac{\color{blue}{y \cdot 0.3333333333333333}}{\sqrt{x}}\right) \]

7. Simplified 99.7%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{y \cdot 0.3333333333333333}{\sqrt{x}}}\right) \]

8. Taylor expanded in x around 0 99.6%

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

9. Final simplification 99.6%

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

Alternative 6: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-/r* 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 99.6% 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 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/r* 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. metadata-eval 99.6%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

5. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 8: 64.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\frac{-3}{y}}{x \cdot \frac{-27}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ (/ -3.0 y) (* x (/ -27.0 y)))))

C

double code(double x, double y) {
	return 1.0 - ((-3.0 / y) / (x * (-27.0 / y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (((-3.0d0) / y) / (x * ((-27.0d0) / y)))
end function

Java

public static double code(double x, double y) {
	return 1.0 - ((-3.0 / y) / (x * (-27.0 / y)));
}

Python

def code(x, y):
	return 1.0 - ((-3.0 / y) / (x * (-27.0 / y)))

Julia

function code(x, y)
	return Float64(1.0 - Float64(Float64(-3.0 / y) / Float64(x * Float64(-27.0 / y))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - ((-3.0 / y) / (x * (-27.0 / y)));
end

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(-3.0 / y), $MachinePrecision] / N[(x * N[(-27.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-/r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}}\right) \]

5. Applied egg-rr 99.7%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333 \cdot \frac{y}{\sqrt{x}}}\right) \]
6. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{0.3333333333333333 \cdot y}{\sqrt{x}}}\right) \]

   2. *-commutative 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \frac{\color{blue}{y \cdot 0.3333333333333333}}{\sqrt{x}}\right) \]

7. Simplified 99.7%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{y \cdot 0.3333333333333333}{\sqrt{x}}}\right) \]
8. Step-by-step derivation

   1. frac-2neg 99.7%

      \[\leadsto 1 - \left(\color{blue}{\frac{-1}{-x \cdot 9}} + \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\right) \]

   2. metadata-eval 99.7%

      \[\leadsto 1 - \left(\frac{\color{blue}{-1}}{-x \cdot 9} + \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\right) \]

   3. metadata-eval 99.7%

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

   4. div-inv 99.8%

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

   5. div-inv 99.7%

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

   6. clear-num 99.7%

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

   7. inv-pow 99.7%

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

   8. inv-pow 99.7%

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

   9. unpow-prod-down 99.7%

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

   10. *-commutative 99.7%

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

   11. inv-pow 99.7%

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

   12. associate-/r* 99.7%

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

   13. frac-add 94.6%

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

   14. pow1/2 94.6%

       \[\leadsto 1 - \frac{-1 \cdot \frac{3}{y} + \left(-x \cdot 9\right) \cdot \frac{1}{\color{blue}{{x}^{0.5}}}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

   15. pow-flip 94.6%

       \[\leadsto 1 - \frac{-1 \cdot \frac{3}{y} + \left(-x \cdot 9\right) \cdot \color{blue}{{x}^{\left(-0.5\right)}}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

   16. metadata-eval 94.6%

       \[\leadsto 1 - \frac{-1 \cdot \frac{3}{y} + \left(-x \cdot 9\right) \cdot {x}^{\color{blue}{-0.5}}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

9. Applied egg-rr 94.6%

   \[\leadsto 1 - \color{blue}{\frac{-1 \cdot \frac{3}{y} + \left(-x \cdot 9\right) \cdot {x}^{-0.5}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}}} \]
10. Step-by-step derivation

    1. *-commutative 94.6%

       \[\leadsto 1 - \frac{\color{blue}{\frac{3}{y} \cdot -1} + \left(-x \cdot 9\right) \cdot {x}^{-0.5}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    2. +-commutative 94.6%

       \[\leadsto 1 - \frac{\color{blue}{\left(-x \cdot 9\right) \cdot {x}^{-0.5} + \frac{3}{y} \cdot -1}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    3. *-commutative 94.6%

       \[\leadsto 1 - \frac{\color{blue}{{x}^{-0.5} \cdot \left(-x \cdot 9\right)} + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    4. distribute-rgt-neg-in 94.6%

       \[\leadsto 1 - \frac{{x}^{-0.5} \cdot \color{blue}{\left(x \cdot \left(-9\right)\right)} + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    5. metadata-eval 94.6%

       \[\leadsto 1 - \frac{{x}^{-0.5} \cdot \left(x \cdot \color{blue}{-9}\right) + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    6. associate-*r* 95.0%

       \[\leadsto 1 - \frac{\color{blue}{\left({x}^{-0.5} \cdot x\right) \cdot -9} + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    7. pow-plus 95.0%

       \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + 1\right)}} \cdot -9 + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    8. metadata-eval 95.0%

       \[\leadsto 1 - \frac{{x}^{\color{blue}{0.5}} \cdot -9 + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    9. associate-*l/ 95.0%

       \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \color{blue}{\frac{3 \cdot -1}{y}}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    10. metadata-eval 95.0%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{\color{blue}{-3}}{y}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    11. associate-*r/ 94.9%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\color{blue}{\frac{\left(-x \cdot 9\right) \cdot 3}{y}}} \]

    12. distribute-rgt-neg-in 94.9%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{\color{blue}{\left(x \cdot \left(-9\right)\right)} \cdot 3}{y}} \]

    13. metadata-eval 94.9%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{\left(x \cdot \color{blue}{-9}\right) \cdot 3}{y}} \]

    14. associate-*l* 95.0%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{\color{blue}{x \cdot \left(-9 \cdot 3\right)}}{y}} \]

    15. metadata-eval 95.0%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{x \cdot \color{blue}{-27}}{y}} \]

11. Simplified 95.0%

    \[\leadsto 1 - \color{blue}{\frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{x \cdot -27}{y}}} \]

12. Taylor expanded in x around 0 94.9%

    \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\color{blue}{-27 \cdot \frac{x}{y}}} \]
13. Step-by-step derivation

    1. associate-*r/ 95.0%

       \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\color{blue}{\frac{-27 \cdot x}{y}}} \]

    2. *-commutative 95.0%

       \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{\color{blue}{x \cdot -27}}{y}} \]

    3. associate-*r/ 95.0%

       \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\color{blue}{x \cdot \frac{-27}{y}}} \]

14. Simplified 95.0%

    \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\color{blue}{x \cdot \frac{-27}{y}}} \]

15. Taylor expanded in x around 0 68.5%

    \[\leadsto 1 - \frac{\color{blue}{\frac{-3}{y}}}{x \cdot \frac{-27}{y}} \]

16. Final simplification 68.5%

    \[\leadsto 1 - \frac{\frac{-3}{y}}{x \cdot \frac{-27}{y}} \]

Alternative 9: 64.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\frac{-3}{y}}{\frac{x \cdot -27}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ (/ -3.0 y) (/ (* x -27.0) y))))

C

double code(double x, double y) {
	return 1.0 - ((-3.0 / y) / ((x * -27.0) / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (((-3.0d0) / y) / ((x * (-27.0d0)) / y))
end function

Java

public static double code(double x, double y) {
	return 1.0 - ((-3.0 / y) / ((x * -27.0) / y));
}

Python

def code(x, y):
	return 1.0 - ((-3.0 / y) / ((x * -27.0) / y))

Julia

function code(x, y)
	return Float64(1.0 - Float64(Float64(-3.0 / y) / Float64(Float64(x * -27.0) / y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - ((-3.0 / y) / ((x * -27.0) / y));
end

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(-3.0 / y), $MachinePrecision] / N[(N[(x * -27.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-/r* 99.8%

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

3. Simplified 99.8%

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

   1. associate-/r* 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. times-frac 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333} \cdot \frac{y}{\sqrt{x}}\right) \]

5. Applied egg-rr 99.7%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{0.3333333333333333 \cdot \frac{y}{\sqrt{x}}}\right) \]
6. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{0.3333333333333333 \cdot y}{\sqrt{x}}}\right) \]

   2. *-commutative 99.7%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \frac{\color{blue}{y \cdot 0.3333333333333333}}{\sqrt{x}}\right) \]

7. Simplified 99.7%

   \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\frac{y \cdot 0.3333333333333333}{\sqrt{x}}}\right) \]
8. Step-by-step derivation

   1. frac-2neg 99.7%

      \[\leadsto 1 - \left(\color{blue}{\frac{-1}{-x \cdot 9}} + \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\right) \]

   2. metadata-eval 99.7%

      \[\leadsto 1 - \left(\frac{\color{blue}{-1}}{-x \cdot 9} + \frac{y \cdot 0.3333333333333333}{\sqrt{x}}\right) \]

   3. metadata-eval 99.7%

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

   4. div-inv 99.8%

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

   5. div-inv 99.7%

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

   6. clear-num 99.7%

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

   7. inv-pow 99.7%

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

   8. inv-pow 99.7%

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

   9. unpow-prod-down 99.7%

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

   10. *-commutative 99.7%

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

   11. inv-pow 99.7%

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

   12. associate-/r* 99.7%

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

   13. frac-add 94.6%

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

   14. pow1/2 94.6%

       \[\leadsto 1 - \frac{-1 \cdot \frac{3}{y} + \left(-x \cdot 9\right) \cdot \frac{1}{\color{blue}{{x}^{0.5}}}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

   15. pow-flip 94.6%

       \[\leadsto 1 - \frac{-1 \cdot \frac{3}{y} + \left(-x \cdot 9\right) \cdot \color{blue}{{x}^{\left(-0.5\right)}}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

   16. metadata-eval 94.6%

       \[\leadsto 1 - \frac{-1 \cdot \frac{3}{y} + \left(-x \cdot 9\right) \cdot {x}^{\color{blue}{-0.5}}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

9. Applied egg-rr 94.6%

   \[\leadsto 1 - \color{blue}{\frac{-1 \cdot \frac{3}{y} + \left(-x \cdot 9\right) \cdot {x}^{-0.5}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}}} \]
10. Step-by-step derivation

    1. *-commutative 94.6%

       \[\leadsto 1 - \frac{\color{blue}{\frac{3}{y} \cdot -1} + \left(-x \cdot 9\right) \cdot {x}^{-0.5}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    2. +-commutative 94.6%

       \[\leadsto 1 - \frac{\color{blue}{\left(-x \cdot 9\right) \cdot {x}^{-0.5} + \frac{3}{y} \cdot -1}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    3. *-commutative 94.6%

       \[\leadsto 1 - \frac{\color{blue}{{x}^{-0.5} \cdot \left(-x \cdot 9\right)} + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    4. distribute-rgt-neg-in 94.6%

       \[\leadsto 1 - \frac{{x}^{-0.5} \cdot \color{blue}{\left(x \cdot \left(-9\right)\right)} + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    5. metadata-eval 94.6%

       \[\leadsto 1 - \frac{{x}^{-0.5} \cdot \left(x \cdot \color{blue}{-9}\right) + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    6. associate-*r* 95.0%

       \[\leadsto 1 - \frac{\color{blue}{\left({x}^{-0.5} \cdot x\right) \cdot -9} + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    7. pow-plus 95.0%

       \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(-0.5 + 1\right)}} \cdot -9 + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    8. metadata-eval 95.0%

       \[\leadsto 1 - \frac{{x}^{\color{blue}{0.5}} \cdot -9 + \frac{3}{y} \cdot -1}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    9. associate-*l/ 95.0%

       \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \color{blue}{\frac{3 \cdot -1}{y}}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    10. metadata-eval 95.0%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{\color{blue}{-3}}{y}}{\left(-x \cdot 9\right) \cdot \frac{3}{y}} \]

    11. associate-*r/ 94.9%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\color{blue}{\frac{\left(-x \cdot 9\right) \cdot 3}{y}}} \]

    12. distribute-rgt-neg-in 94.9%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{\color{blue}{\left(x \cdot \left(-9\right)\right)} \cdot 3}{y}} \]

    13. metadata-eval 94.9%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{\left(x \cdot \color{blue}{-9}\right) \cdot 3}{y}} \]

    14. associate-*l* 95.0%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{\color{blue}{x \cdot \left(-9 \cdot 3\right)}}{y}} \]

    15. metadata-eval 95.0%

        \[\leadsto 1 - \frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{x \cdot \color{blue}{-27}}{y}} \]

11. Simplified 95.0%

    \[\leadsto 1 - \color{blue}{\frac{{x}^{0.5} \cdot -9 + \frac{-3}{y}}{\frac{x \cdot -27}{y}}} \]

12. Taylor expanded in x around 0 68.6%

    \[\leadsto 1 - \frac{\color{blue}{\frac{-3}{y}}}{\frac{x \cdot -27}{y}} \]

13. Final simplification 68.6%

    \[\leadsto 1 - \frac{\frac{-3}{y}}{\frac{x \cdot -27}{y}} \]

Alternative 10: 63.2% 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 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/r* 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 67.7%

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

   1. associate-*r/ 67.7%

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

   2. metadata-eval 67.7%

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

6. Simplified 67.7%

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

   1. div-inv 67.7%

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

   2. metadata-eval 67.7%

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

   3. inv-pow 67.7%

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

   4. unpow-prod-down 67.9%

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

   5. *-commutative 67.9%

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

   6. unpow-prod-down 67.7%

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

   7. inv-pow 67.7%

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

   8. metadata-eval 67.7%

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

8. Applied egg-rr 67.7%

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

   1. associate-*l/ 67.7%

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

   2. metadata-eval 67.7%

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

   3. clear-num 67.8%

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

   4. div-inv 67.9%

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

   5. metadata-eval 67.9%

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

10. Applied egg-rr 67.9%

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

11. Final simplification 67.9%

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

Alternative 11: 63.1% 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 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/r* 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 67.7%

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

   1. associate-*r/ 67.7%

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

   2. metadata-eval 67.7%

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

6. Simplified 67.7%

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

7. Final simplification 67.7%

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

Alternative 12: 32.1% 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 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-/r* 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around inf 35.7%

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

5. Final simplification 35.7%

   \[\leadsto 1 \]

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 2023199 
(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)))))