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

Percentage Accurate: 99.7% → 99.7%

Time: 7.0s

Alternatives: 14

Speedup: 0.3×

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, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

   \[\leadsto \left(1 - {\left(x \cdot 9\right)}^{-1}\right) - \frac{\sqrt{{x}^{-1}} \cdot y}{3} \]
9. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 - pow((x * 9.0), -1.0)) - ((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 - ((x * 9.0d0) ** (-1.0d0))) - ((y / 3.0d0) / sqrt(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

   \[\leadsto \left(1 - {\left(x \cdot 9\right)}^{-1}\right) - \frac{\frac{y}{3}}{\sqrt{x}} \]
6. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 - pow((x * 9.0), -1.0)) - (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 - ((x * 9.0d0) ** (-1.0d0))) - (y / (3.0d0 * sqrt(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $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. Add Preprocessing

3. Final simplification 99.7%

   \[\leadsto \left(1 - {\left(x \cdot 9\right)}^{-1}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
4. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 80000000000000:\\ \;\;\;\;\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\sqrt{{x}^{-1}} \cdot y}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 80000000000000.0)
   (/ (+ -0.1111111111111111 (fma (* -0.3333333333333333 (sqrt x)) y x)) x)
   (- 1.0 (/ (* (sqrt (pow x -1.0)) y) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 80000000000000.0) {
		tmp = (-0.1111111111111111 + fma((-0.3333333333333333 * sqrt(x)), y, x)) / x;
	} else {
		tmp = 1.0 - ((sqrt(pow(x, -1.0)) * y) / 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 80000000000000.0)
		tmp = Float64(Float64(-0.1111111111111111 + fma(Float64(-0.3333333333333333 * sqrt(x)), y, x)) / x);
	else
		tmp = Float64(1.0 - Float64(Float64(sqrt((x ^ -1.0)) * y) / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 80000000000000.0], N[(N[(-0.1111111111111111 + N[(N[(-0.3333333333333333 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e13

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. associate--r+ N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. mul-1-neg N/A

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

   7. Applied rewrites 99.4%

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

   if 8e13 < x

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{\sqrt{\frac{1}{x}} \cdot y}{3} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 80000000000000:\\ \;\;\;\;\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\sqrt{{x}^{-1}} \cdot y}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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) * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 6: 94.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+45} \lor \neg \left(y \leq 4.6 \cdot 10^{+53}\right):\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.1e+45) (not (<= y 4.6e+53)))
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (/ (- x 0.1111111111111111) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+45) || !(y <= 4.6e+53)) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else {
		tmp = (x - 0.1111111111111111) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.1d+45)) .or. (.not. (y <= 4.6d+53))) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else
        tmp = (x - 0.1111111111111111d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+45) || !(y <= 4.6e+53)) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else {
		tmp = (x - 0.1111111111111111) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.1e+45) or not (y <= 4.6e+53):
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	else:
		tmp = (x - 0.1111111111111111) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.1e+45) || !(y <= 4.6e+53))
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	else
		tmp = Float64(Float64(x - 0.1111111111111111) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.1e+45) || ~((y <= 4.6e+53)))
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	else
		tmp = (x - 0.1111111111111111) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.1e+45], N[Not[LessEqual[y, 4.6e+53]], $MachinePrecision]], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.09999999999999995e45 or 4.60000000000000039e53 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 94.2%

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

   if -2.09999999999999995e45 < y < 4.60000000000000039e53

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 99.6

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in y around 0

       \[\leadsto \frac{x - \frac{1}{9}}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 98.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+45} \lor \neg \left(y \leq 4.6 \cdot 10^{+53}\right):\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 80000000000000:\\ \;\;\;\;\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 80000000000000.0)
   (/ (+ -0.1111111111111111 (fma (* -0.3333333333333333 (sqrt x)) y x)) x)
   (- 1.0 (/ (/ y 3.0) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 80000000000000.0) {
		tmp = (-0.1111111111111111 + fma((-0.3333333333333333 * sqrt(x)), y, x)) / x;
	} else {
		tmp = 1.0 - ((y / 3.0) / sqrt(x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 80000000000000.0)
		tmp = Float64(Float64(-0.1111111111111111 + fma(Float64(-0.3333333333333333 * sqrt(x)), y, x)) / x);
	else
		tmp = Float64(1.0 - Float64(Float64(y / 3.0) / sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 80000000000000.0], N[(N[(-0.1111111111111111 + N[(N[(-0.3333333333333333 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e13

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. associate--r+ N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. mul-1-neg N/A

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

   7. Applied rewrites 99.4%

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

   if 8e13 < x

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} - \frac{\frac{y}{3}}{\sqrt{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 80000000000000:\\ \;\;\;\;\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e+52)
   (/ (+ -0.1111111111111111 (fma (* -0.3333333333333333 (sqrt x)) y x)) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e+52) {
		tmp = (-0.1111111111111111 + fma((-0.3333333333333333 * sqrt(x)), y, x)) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e+52)
		tmp = Float64(Float64(-0.1111111111111111 + fma(Float64(-0.3333333333333333 * sqrt(x)), y, x)) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e+52], N[(N[(-0.1111111111111111 + N[(N[(-0.3333333333333333 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e52

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. associate--r+ N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. mul-1-neg N/A

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

   7. Applied rewrites 99.5%

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

   if 5e52 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{-0.1111111111111111 + \mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e+26)
   (/ (- x (fma (* (sqrt x) y) 0.3333333333333333 0.1111111111111111)) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e+26) {
		tmp = (x - fma((sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e+26)
		tmp = Float64(Float64(x - fma(Float64(sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e+26], N[(N[(x - N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 0.3333333333333333 + 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e26

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 5.0000000000000001e26 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate--l- N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 99.7

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

   16. lift-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-/r* N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. lower-/.f64 N/A

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

   23. metadata-eval 99.6

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

6. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate-/r* N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. associate--r+ N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-sqrt.f64 N/A

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

   14. associate--l- N/A

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

   15. lift-sqrt.f64 N/A

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

   16. associate-/r* N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lower--.f64 N/A

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

8. Applied rewrites 99.5%

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

Alternative 11: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000385:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.000385)
   (/ (fma (* -0.3333333333333333 (sqrt x)) y -0.1111111111111111) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.000385) {
		tmp = fma((-0.3333333333333333 * sqrt(x)), y, -0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.000385)
		tmp = Float64(fma(Float64(-0.3333333333333333 * sqrt(x)), y, -0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.000385], N[(N[(N[(-0.3333333333333333 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8499999999999998e-4

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 97.3%

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

   if 3.8499999999999998e-4 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000385:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot \sqrt{x}, y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000385:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.000385)
   (/ (fma -0.3333333333333333 (* (sqrt x) y) -0.1111111111111111) x)
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.000385) {
		tmp = fma(-0.3333333333333333, (sqrt(x) * y), -0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.000385)
		tmp = Float64(fma(-0.3333333333333333, Float64(sqrt(x) * y), -0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.000385], N[(N[(-0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8499999999999998e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 3.8499999999999998e-4 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 61.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 92.8

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

10. Applied rewrites 92.8%

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

11. Taylor expanded in y around 0

    \[\leadsto \frac{x - \frac{1}{9}}{x} \]
12. Step-by-step derivation

13. Applied rewrites 61.7%

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

14. Final simplification 61.7%

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

Alternative 14: 30.7% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 67.2

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

5. Applied rewrites 67.2%

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

6. Taylor expanded in y around 0

   \[\leadsto \frac{\frac{-1}{9}}{x} \]
7. Step-by-step derivation

8. Applied rewrites 36.4%

   \[\leadsto \frac{-0.1111111111111111}{x} \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.9× 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 2024326 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x)))))

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