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

Percentage Accurate: 99.7% → 99.7%

Time: 5.7s

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, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(N[(-1.0 / x), $MachinePrecision] / -9.0), $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. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. lower-/.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(9.0 * x), $MachinePrecision]), $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. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. lift-/.f64 N/A

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

   7. inv-pow N/A

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

   8. lift-*.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. inv-pow N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. neg-mul-1 N/A

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

   14. un-div-inv N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower--.f64 99.6

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+19}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{x} \cdot y, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e+19)
   (- 1.0 (/ (fma 0.3333333333333333 (* (sqrt x) y) 0.1111111111111111) x))
   (- 1.0 (/ y (* (sqrt x) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e+19) {
		tmp = 1.0 - (fma(0.3333333333333333, (sqrt(x) * y), 0.1111111111111111) / x);
	} else {
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e+19)
		tmp = Float64(1.0 - Float64(fma(0.3333333333333333, Float64(sqrt(x) * y), 0.1111111111111111) / x));
	else
		tmp = Float64(1.0 - Float64(y / Float64(sqrt(x) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e+19], N[(1.0 - N[(N[(0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e19

   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 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%

      \[\leadsto \color{blue}{\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.4%

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

   if 2e19 < x

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+19}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{x} \cdot y, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-mul-1 N/A

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

   7. lift-*.f64 N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 99.6

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

   16. lift-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-/r* N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. lower-/.f64 N/A

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

   23. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

Alternative 6: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+63}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(0.1111111111111111, \frac{-1}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e+63)
   (- 1.0 (/ y (* (sqrt x) 3.0)))
   (if (<= y 6.5e+43)
     (fma 0.1111111111111111 (/ -1.0 x) 1.0)
     (fma (/ y (sqrt x)) -0.3333333333333333 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+63) {
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	} else if (y <= 6.5e+43) {
		tmp = fma(0.1111111111111111, (-1.0 / x), 1.0);
	} else {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e+63)
		tmp = Float64(1.0 - Float64(y / Float64(sqrt(x) * 3.0)));
	elseif (y <= 6.5e+43)
		tmp = fma(0.1111111111111111, Float64(-1.0 / x), 1.0);
	else
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e+63], N[(1.0 - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+43], N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0500000000000001e63

   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 95.4%

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

   if -1.0500000000000001e63 < y < 6.4999999999999998e43

   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

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.2%

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

   10. Applied rewrites 98.2%

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

   12. Applied rewrites 98.3%

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

   if 6.4999999999999998e43 < y

   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 inf

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

   5. Applied rewrites 94.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   7. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+63}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(0.1111111111111111, \frac{-1}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(0.1111111111111111, \frac{-1}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e+63)
   (fma (/ -0.3333333333333333 (sqrt x)) y 1.0)
   (if (<= y 6.5e+43)
     (fma 0.1111111111111111 (/ -1.0 x) 1.0)
     (fma (/ y (sqrt x)) -0.3333333333333333 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+63) {
		tmp = fma((-0.3333333333333333 / sqrt(x)), y, 1.0);
	} else if (y <= 6.5e+43) {
		tmp = fma(0.1111111111111111, (-1.0 / x), 1.0);
	} else {
		tmp = fma((y / sqrt(x)), -0.3333333333333333, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e+63)
		tmp = fma(Float64(-0.3333333333333333 / sqrt(x)), y, 1.0);
	elseif (y <= 6.5e+43)
		tmp = fma(0.1111111111111111, Float64(-1.0 / x), 1.0);
	else
		tmp = fma(Float64(y / sqrt(x)), -0.3333333333333333, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e+63], N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 6.5e+43], N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0500000000000001e63

   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 95.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-fma.f64 95.3

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

   7. Applied rewrites 95.3%

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

   if -1.0500000000000001e63 < y < 6.4999999999999998e43

   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

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.2%

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

   10. Applied rewrites 98.2%

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

   12. Applied rewrites 98.3%

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

   if 6.4999999999999998e43 < y

   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 inf

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

   5. Applied rewrites 94.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

   7. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\sqrt{x}}, -0.3333333333333333, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(0.1111111111111111, \frac{-1}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ -0.3333333333333333 (sqrt x)) y 1.0)))
   (if (<= y -1.05e+63)
     t_0
     (if (<= y 6.5e+43) (fma 0.1111111111111111 (/ -1.0 x) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((-0.3333333333333333 / sqrt(x)), y, 1.0);
	double tmp;
	if (y <= -1.05e+63) {
		tmp = t_0;
	} else if (y <= 6.5e+43) {
		tmp = fma(0.1111111111111111, (-1.0 / x), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(-0.3333333333333333 / sqrt(x)), y, 1.0)
	tmp = 0.0
	if (y <= -1.05e+63)
		tmp = t_0;
	elseif (y <= 6.5e+43)
		tmp = fma(0.1111111111111111, Float64(-1.0 / x), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision]}, If[LessEqual[y, -1.05e+63], t$95$0, If[LessEqual[y, 6.5e+43], N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0500000000000001e63 or 6.4999999999999998e43 < y

   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 inf

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

   5. Applied rewrites 95.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-fma.f64 95.1

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

   7. Applied rewrites 95.1%

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

   if -1.0500000000000001e63 < y < 6.4999999999999998e43

   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

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.2%

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

   10. Applied rewrites 98.2%

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

   12. Applied rewrites 98.3%

       \[\leadsto \mathsf{fma}\left(0.1111111111111111, \color{blue}{\frac{-1}{x}}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (fma -0.3333333333333333 (* (sqrt x) y) -0.1111111111111111) x)
   (- 1.0 (/ y (* (sqrt x) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = fma(-0.3333333333333333, (sqrt(x) * y), -0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(fma(-0.3333333333333333, Float64(sqrt(x) * y), -0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(sqrt(x) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(-0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

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

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. distribute-neg-frac N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.5

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-/r* N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 99.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(0.1111111111111111, \frac{-1}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111 \cdot x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e+154)
   (fma 0.1111111111111111 (/ -1.0 x) 1.0)
   (- 1.0 (/ (* 0.1111111111111111 x) (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.4e+154) {
		tmp = fma(0.1111111111111111, (-1.0 / x), 1.0);
	} else {
		tmp = 1.0 - ((0.1111111111111111 * x) / (x * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.4e+154)
		tmp = fma(0.1111111111111111, Float64(-1.0 / x), 1.0);
	else
		tmp = Float64(1.0 - Float64(Float64(0.1111111111111111 * x) / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.4e+154], N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 - N[(N[(0.1111111111111111 * x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4e154

   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 96.6

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.3%

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

   10. Applied rewrites 71.3%

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

   12. Applied rewrites 71.4%

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

   if 1.4e154 < y

   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

      \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 4.2%

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

   10. Applied rewrites 23.5%

       \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot x}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{\frac{x - \left(\frac{1}{9} + \frac{1}{3} \cdot \left(\sqrt{x} \cdot y\right)\right)}{x}} \]
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 95.5

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

5. Applied rewrites 95.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 62.4%

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

10. Applied rewrites 62.4%

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

12. Applied rewrites 62.4%

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

Alternative 12: 62.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. 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 95.5

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

5. Applied rewrites 95.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 62.4%

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

10. Applied rewrites 62.4%

    \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]
11. 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 2024298 
(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)))))