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

Percentage Accurate: 99.7% → 99.7%

Time: 9.2s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 62.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \leq -50:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ 1.0 (/ -1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))) -50.0)
   (/ -0.1111111111111111 x)
   1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 + (-1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)))) <= -50.0) {
		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) / (x * 9.0d0))) - (y / (3.0d0 * sqrt(x)))) <= (-50.0d0)) 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 / (x * 9.0))) - (y / (3.0 * Math.sqrt(x)))) <= -50.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 + (-1.0 / (x * 9.0))) - (y / (3.0 * math.sqrt(x)))) <= -50.0:
		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(x * 9.0))) - Float64(y / Float64(3.0 * sqrt(x)))) <= -50.0)
		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 / (x * 9.0))) - (y / (3.0 * sqrt(x)))) <= -50.0)
		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[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -50.0], 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)))) < -50

   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 y around 0

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 61.4

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

   5. Simplified 61.4%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 61.4

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

   8. Simplified 61.4%

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

   if -50 < (-.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. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 65.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \leq -50:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. inv-pow N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

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

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. --lowering--.f64 N/A

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

   13. /-lowering-/.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 4: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+42}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y (* 3.0 (sqrt x))))))
   (if (<= y -5.5e+86) t_0 (if (<= y 8e+42) (+ 1.0 (/ 1.0 (* x -9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (y / (3.0 * sqrt(x)));
	double tmp;
	if (y <= -5.5e+86) {
		tmp = t_0;
	} else if (y <= 8e+42) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    if (y <= (-5.5d+86)) then
        tmp = t_0
    else if (y <= 8d+42) then
        tmp = 1.0d0 + (1.0d0 / (x * (-9.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (y / (3.0 * Math.sqrt(x)));
	double tmp;
	if (y <= -5.5e+86) {
		tmp = t_0;
	} else if (y <= 8e+42) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (y / (3.0 * math.sqrt(x)))
	tmp = 0
	if y <= -5.5e+86:
		tmp = t_0
	elif y <= 8e+42:
		tmp = 1.0 + (1.0 / (x * -9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))))
	tmp = 0.0
	if (y <= -5.5e+86)
		tmp = t_0;
	elseif (y <= 8e+42)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (y / (3.0 * sqrt(x)));
	tmp = 0.0;
	if (y <= -5.5e+86)
		tmp = t_0;
	elseif (y <= 8e+42)
		tmp = 1.0 + (1.0 / (x * -9.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+86], t$95$0, If[LessEqual[y, 8e+42], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000002e86 or 8.00000000000000036e42 < 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. Simplified 97.8%

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

   if -5.5000000000000002e86 < y < 8.00000000000000036e42

   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. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 98.6

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

   5. Simplified 98.6%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval 98.6

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

   7. Applied egg-rr 98.6%

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

Alternative 5: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

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

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

   6. distribute-frac-neg2 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. distribute-frac-neg2 N/A

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

   9. associate-/r* N/A

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

   10. distribute-neg-frac N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. sqrt-lowering-sqrt.f64 N/A

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

   15. sub-neg N/A

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

   16. +-lowering-+.f64 N/A

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

   17. *-commutative N/A

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

   18. associate-/r* N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

4. Applied egg-rr 99.7%

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

Alternative 6: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, 1\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+44}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ -0.3333333333333333 (sqrt x)) y 1.0)))
   (if (<= y -6e+86) t_0 (if (<= y 8.5e+44) (+ 1.0 (/ 1.0 (* x -9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((-0.3333333333333333 / sqrt(x)), y, 1.0);
	double tmp;
	if (y <= -6e+86) {
		tmp = t_0;
	} else if (y <= 8.5e+44) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} 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 <= -6e+86)
		tmp = t_0;
	elseif (y <= 8.5e+44)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	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, -6e+86], t$95$0, If[LessEqual[y, 8.5e+44], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.99999999999999954e86 or 8.5e44 < 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

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3} \cdot y\right)\right) + 0}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot y} + 0, 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \color{blue}{\frac{-1}{3}} \cdot y + 0, 1\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \color{blue}{y \cdot \frac{-1}{3}} + 0, 1\right) \]

      16. accelerator-lowering-fma.f64 97.6

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \color{blue}{\mathsf{fma}\left(y, -0.3333333333333333, 0\right)}, 1\right) \]

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, \mathsf{fma}\left(y, -0.3333333333333333, 0\right), 1\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

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

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sqrt-lowering-sqrt.f64 97.7

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

   7. Applied egg-rr 97.7%

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

   if -5.99999999999999954e86 < y < 8.5e44

   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. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 98.6

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

   5. Simplified 98.6%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval 98.6

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

   7. Applied egg-rr 98.6%

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

Alternative 7: 98.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (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 <= 0.11) {
		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 <= 0.11)
		tmp = Float64(fma(sqrt(x), Float64(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, 0.11], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision] + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 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. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. div-inv N/A

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

      4. *-lowering-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-lowering-sqrt.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0

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

      1. 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. /-lowering-/.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. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. sqrt-lowering-sqrt.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 98.8

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

   7. Simplified 98.8%

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

   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. Simplified 99.4%

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

Alternative 8: 63.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. 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. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. /-lowering-/.f64 63.7

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

5. Simplified 63.7%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

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

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

   6. /-lowering-/.f64 N/A

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

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

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

   8. *-lowering-*.f64 N/A

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

   9. metadata-eval 63.7

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

7. Applied egg-rr 63.7%

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

Alternative 9: 63.2% 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. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. /-lowering-/.f64 63.7

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

5. Simplified 63.7%

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

Alternative 10: 32.0% 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. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. /-lowering-/.f64 63.7

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

5. Simplified 63.7%

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

6. Taylor expanded in x around inf

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

8. Simplified 35.0%

   \[\leadsto \color{blue}{1} \]
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 2024195 
(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)))))