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

Percentage Accurate: 99.7% → 99.7%

Time: 10.0s

Alternatives: 14

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \leq -0.2:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ 1.0 (/ -1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))) -0.2)
   (/ -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)))) <= -0.2) {
		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)))) <= (-0.2d0)) 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)))) <= -0.2) {
		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)))) <= -0.2:
		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)))) <= -0.2)
		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)))) <= -0.2)
		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], -0.2], 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)))) < -0.20000000000000001

   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 68.1

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

   5. Simplified 68.1%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 66.1

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

   8. Simplified 66.1%

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

   if -0.20000000000000001 < (-.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 63.1

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

   5. Simplified 63.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 60.9%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \leq -0.2:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
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 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(-1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(y * 0.3333333333333333), $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. 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. distribute-neg-frac N/A

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

   4. neg-mul-1 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

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

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

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

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

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

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

   10. div-inv N/A

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

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

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

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

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

   15. *-commutative N/A

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

   16. associate-/r* N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. distribute-neg-frac N/A

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

   20. /-lowering-/.f64 N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 5: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0285:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, y \cdot -0.3333333333333333, -0.1111111111111111\right)}{x}\\ \mathbf{elif}\;x \leq 25000000:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0285)
   (/ (fma (sqrt x) (* y -0.3333333333333333) -0.1111111111111111) x)
   (if (<= x 25000000.0)
     (+ 1.0 (/ -0.1111111111111111 x))
     (- 1.0 (/ y (* 3.0 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0285) {
		tmp = fma(sqrt(x), (y * -0.3333333333333333), -0.1111111111111111) / x;
	} else if (x <= 25000000.0) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0285)
		tmp = Float64(fma(sqrt(x), Float64(y * -0.3333333333333333), -0.1111111111111111) / x);
	elseif (x <= 25000000.0)
		tmp = Float64(1.0 + Float64(-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, 0.0285], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision] + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 25000000.0], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.028500000000000001

   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

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

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

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

      3. /-lowering-/.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   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. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      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 97.7

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

   7. Simplified 97.7%

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

   if 0.028500000000000001 < x < 2.5e7

   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 99.8

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

   5. Simplified 99.8%

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

   if 2.5e7 < 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 99.8%

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

Alternative 6: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y (* 3.0 (sqrt x))))))
   (if (<= y -1.9e+32)
     t_0
     (if (<= y 1.2e+81) (+ 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 <= -1.9e+32) {
		tmp = t_0;
	} else if (y <= 1.2e+81) {
		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 <= (-1.9d+32)) then
        tmp = t_0
    else if (y <= 1.2d+81) 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 <= -1.9e+32) {
		tmp = t_0;
	} else if (y <= 1.2e+81) {
		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 <= -1.9e+32:
		tmp = t_0
	elif y <= 1.2e+81:
		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 <= -1.9e+32)
		tmp = t_0;
	elseif (y <= 1.2e+81)
		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 <= -1.9e+32)
		tmp = t_0;
	elseif (y <= 1.2e+81)
		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, -1.9e+32], t$95$0, If[LessEqual[y, 1.2e+81], 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 < -1.9000000000000002e32 or 1.19999999999999995e81 < 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 93.1%

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

   if -1.9000000000000002e32 < y < 1.19999999999999995e81

   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 98.0

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

   5. Simplified 98.0%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval 98.1

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

   7. Applied egg-rr 98.1%

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

Alternative 7: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(\sqrt{x} \cdot 0.3333333333333333, y, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.4e+15)
   (- 1.0 (/ (fma (* (sqrt x) 0.3333333333333333) y 0.1111111111111111) x))
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.4e+15) {
		tmp = 1.0 - (fma((sqrt(x) * 0.3333333333333333), y, 0.1111111111111111) / x);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.4e+15)
		tmp = Float64(1.0 - Float64(fma(Float64(sqrt(x) * 0.3333333333333333), y, 0.1111111111111111) / x));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.4e+15], N[(1.0 - N[(N[(N[(N[Sqrt[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y + 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $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 < 2.4e15

   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

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

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

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

      3. /-lowering-/.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. --lowering--.f64 N/A

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 99.4

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

   7. Simplified 99.4%

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. sqrt-lowering-sqrt.f64 99.5

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

   9. Applied egg-rr 99.5%

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

   if 2.4e15 < 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 99.8%

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

Alternative 8: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(0.3333333333333333, y \cdot \sqrt{x}, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e+15)
   (- 1.0 (/ (fma 0.3333333333333333 (* y (sqrt x)) 0.1111111111111111) x))
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e+15) {
		tmp = 1.0 - (fma(0.3333333333333333, (y * sqrt(x)), 0.1111111111111111) / x);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e+15)
		tmp = Float64(1.0 - Float64(fma(0.3333333333333333, Float64(y * sqrt(x)), 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, 2e+15], N[(1.0 - N[(N[(0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $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 < 2e15

   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

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

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

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

      3. /-lowering-/.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. --lowering--.f64 N/A

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 99.4

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

   7. Simplified 99.4%

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

   if 2e15 < 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 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(0.3333333333333333, y \cdot \sqrt{x}, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+31}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e+31)
   (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x)))
   (if (<= y 1.2e+81)
     (+ 1.0 (/ 1.0 (* x -9.0)))
     (fma y (/ -0.3333333333333333 (sqrt x)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e+31) {
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	} else if (y <= 1.2e+81) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = fma(y, (-0.3333333333333333 / sqrt(x)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e+31)
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	elseif (y <= 1.2e+81)
		tmp = Float64(1.0 + Float64(1.0 / Float64(x * -9.0)));
	else
		tmp = fma(y, Float64(-0.3333333333333333 / sqrt(x)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e+31], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+81], N[(1.0 + N[(1.0 / N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.9999999999999992e31

   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 90.5

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

   5. Simplified 90.5%

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

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

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

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

      8. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{-1}{3} + 0}}{\sqrt{x}} + 1 \]

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

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

      10. sqrt-lowering-sqrt.f64 90.6

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

   7. Applied egg-rr 90.6%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 90.6

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

   9. Applied egg-rr 90.6%

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

   if -8.9999999999999992e31 < y < 1.19999999999999995e81

   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 98.0

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

   5. Simplified 98.0%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval 98.1

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

   7. Applied egg-rr 98.1%

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

   if 1.19999999999999995e81 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 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 95.9

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

   5. Simplified 95.9%

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

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

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

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

      8. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{-1}{3} + 0}}{\sqrt{x}} + 1 \]

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

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

      10. sqrt-lowering-sqrt.f64 96.1

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

   7. Applied egg-rr 96.1%

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

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 96.2

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

   9. Applied egg-rr 96.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+31}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, 1\right)\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma y (/ -0.3333333333333333 (sqrt x)) 1.0)))
   (if (<= y -8.5e+34)
     t_0
     (if (<= y 1.2e+81) (+ 1.0 (/ 1.0 (* x -9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, (-0.3333333333333333 / sqrt(x)), 1.0);
	double tmp;
	if (y <= -8.5e+34) {
		tmp = t_0;
	} else if (y <= 1.2e+81) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, Float64(-0.3333333333333333 / sqrt(x)), 1.0)
	tmp = 0.0
	if (y <= -8.5e+34)
		tmp = t_0;
	elseif (y <= 1.2e+81)
		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[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -8.5e+34], t$95$0, If[LessEqual[y, 1.2e+81], 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 < -8.5000000000000003e34 or 1.19999999999999995e81 < 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 92.9

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

   5. Simplified 92.9%

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

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

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

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

      8. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{-1}{3} + 0}}{\sqrt{x}} + 1 \]

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

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

      10. sqrt-lowering-sqrt.f64 93.0

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

   7. Applied egg-rr 93.0%

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

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 93.1

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

   9. Applied egg-rr 93.1%

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

   if -8.5000000000000003e34 < y < 1.19999999999999995e81

   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 98.0

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

   5. Simplified 98.0%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval 98.1

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

   7. Applied egg-rr 98.1%

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

Alternative 11: 94.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1\right)\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{1}{x \cdot -9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.3333333333333333 (/ y (sqrt x)) 1.0)))
   (if (<= y -1.15e+32)
     t_0
     (if (<= y 1.2e+81) (+ 1.0 (/ 1.0 (* x -9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(-0.3333333333333333, (y / sqrt(x)), 1.0);
	double tmp;
	if (y <= -1.15e+32) {
		tmp = t_0;
	} else if (y <= 1.2e+81) {
		tmp = 1.0 + (1.0 / (x * -9.0));
	} 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 <= -1.15e+32)
		tmp = t_0;
	elseif (y <= 1.2e+81)
		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[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -1.15e+32], t$95$0, If[LessEqual[y, 1.2e+81], 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 < -1.15e32 or 1.19999999999999995e81 < 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 92.9

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

   5. Simplified 92.9%

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

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

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

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

      8. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{-1}{3} + 0}}{\sqrt{x}} + 1 \]

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

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

      10. sqrt-lowering-sqrt.f64 93.0

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

   7. Applied egg-rr 93.0%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 93.0

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

   9. Applied egg-rr 93.0%

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

   if -1.15e32 < y < 1.19999999999999995e81

   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 98.0

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

   5. Simplified 98.0%

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

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval 98.1

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

   7. Applied egg-rr 98.1%

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

Alternative 12: 63.4% 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 65.7

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

5. Simplified 65.7%

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

   1. clear-num N/A

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

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

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. metadata-eval 65.8

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

7. Applied egg-rr 65.8%

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

Alternative 13: 63.4% 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 65.7

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

5. Simplified 65.7%

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

Alternative 14: 31.4% 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 65.7

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

5. Simplified 65.7%

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

6. Taylor expanded in x around inf

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

8. Simplified 30.3%

   \[\leadsto \color{blue}{1} \]
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 2024196 
(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)))))