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

Percentage Accurate: 99.7% → 99.7%

Time: 6.6s

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

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

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(Float64(y / 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[(N[(y / N[Sqrt[x], $MachinePrecision]), $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. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. 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. Final simplification 99.7%

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

Alternative 2: 61.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{1}{9 \cdot x}\right) - \frac{y}{3 \cdot \sqrt{x}} \leq -1000000000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (- 1.0 (/ 1.0 (* 9.0 x))) (/ y (* 3.0 (sqrt x)))) -1000000000.0)
   (/ -0.1111111111111111 x)
   1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - (1.0 / (9.0 * x))) - (y / (3.0 * sqrt(x)))) <= -1000000000.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((1.0d0 - (1.0d0 / (9.0d0 * x))) - (y / (3.0d0 * sqrt(x)))) <= (-1000000000.0d0)) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - (1.0 / (9.0 * x))) - (y / (3.0 * Math.sqrt(x)))) <= -1000000000.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - (1.0 / (9.0 * x))) - (y / (3.0 * math.sqrt(x)))) <= -1000000000.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(1.0 / Float64(9.0 * x))) - Float64(y / Float64(3.0 * sqrt(x)))) <= -1000000000.0)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - (1.0 / (9.0 * x))) - (y / (3.0 * sqrt(x)))) <= -1000000000.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1000000000.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) (/.f64 y (*.f64 #s(literal 3 binary64) (sqrt.f64 x)))) < -1e9

   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

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.3%

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

   if -1e9 < (-.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (*.f64 x #s(literal 9 binary64)))) (/.f64 y (*.f64 #s(literal 3 binary64) (sqrt.f64 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 y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 60.2%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{1}{9 \cdot x}\right) - \frac{y}{3 \cdot \sqrt{x}} \leq -1000000000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(-0.3333333333333333 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * y), $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. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+15}:\\ \;\;\;\;\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.02e+15)
   (/ (- 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 <= 1.02e+15) {
		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 <= 1.02e+15)
		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, 1.02e+15], 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 < 1.02e15

   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%

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

   if 1.02e15 < x

   1. Initial program 99.9%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 99.9%

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

Alternative 6: 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 7: 95.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+47}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+46}:\\ \;\;\;\;1 - \frac{1}{9 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e+47)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 1.06e+46)
     (- 1.0 (/ 1.0 (* 9.0 x)))
     (fma -0.3333333333333333 (/ y (sqrt x)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+47) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 1.06e+46) {
		tmp = 1.0 - (1.0 / (9.0 * x));
	} else {
		tmp = fma(-0.3333333333333333, (y / sqrt(x)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5e+47)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 1.06e+46)
		tmp = Float64(1.0 - Float64(1.0 / Float64(9.0 * x)));
	else
		tmp = fma(-0.3333333333333333, Float64(y / sqrt(x)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5e+47], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e+46], N[(1.0 - N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5000000000000008e47

   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 97.3%

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

   if -8.5000000000000008e47 < y < 1.05999999999999998e46

   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 y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.6%

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

   if 1.05999999999999998e46 < y

   1. Initial program 99.5%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 99.5

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 96.2%

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

Alternative 8: 95.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1\right)\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+46}:\\ \;\;\;\;1 - \frac{1}{9 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.3333333333333333 (/ y (sqrt x)) 1.0)))
   (if (<= y -8.5e+47)
     t_0
     (if (<= y 1.06e+46) (- 1.0 (/ 1.0 (* 9.0 x))) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(-0.3333333333333333, (y / sqrt(x)), 1.0);
	double tmp;
	if (y <= -8.5e+47) {
		tmp = t_0;
	} else if (y <= 1.06e+46) {
		tmp = 1.0 - (1.0 / (9.0 * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.3333333333333333, Float64(y / sqrt(x)), 1.0)
	tmp = 0.0
	if (y <= -8.5e+47)
		tmp = t_0;
	elseif (y <= 1.06e+46)
		tmp = Float64(1.0 - Float64(1.0 / Float64(9.0 * x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -8.5e+47], t$95$0, If[LessEqual[y, 1.06e+46], N[(1.0 - N[(1.0 / N[(9.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5000000000000008e47 or 1.05999999999999998e46 < y

   1. Initial program 99.5%

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

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 96.7%

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

   if -8.5000000000000008e47 < y < 1.05999999999999998e46

   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 y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.6%

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

Alternative 9: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00037:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x} \cdot y, -0.3333333333333333, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00037)
   (/ (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 <= 0.00037) {
		tmp = 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 <= 0.00037)
		tmp = Float64(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, 0.00037], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * -0.3333333333333333 + -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 < 3.6999999999999999e-4

   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

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 3.6999999999999999e-4 < 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 98.9%

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

Alternative 10: 62.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 - N[(1.0 / N[(9.0 * 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. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 60.7

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

5. Applied rewrites 60.7%

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

7. Applied rewrites 60.8%

   \[\leadsto 1 - \frac{1}{\color{blue}{9 \cdot x}} \]
8. Add Preprocessing

Alternative 11: 62.7% 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 y around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 60.7

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

5. Applied rewrites 60.7%

   \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
6. Add Preprocessing

Alternative 12: 31.6% accurate, 49.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 60.7

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

5. Applied rewrites 60.7%

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

6. Taylor expanded in x around inf

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 32.5%

   \[\leadsto 1 \]
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 2024277 
(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)))))