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

Percentage Accurate: 99.7% → 99.6%

Time: 9.8s

Alternatives: 8

Speedup: N/A×

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: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

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

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

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

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

   14. sub-neg N/A

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

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

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

   16. *-commutative N/A

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

   17. associate-/r* N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. distribute-neg-frac N/A

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

   21. /-lowering-/.f64 N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 2: 61.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot 3} \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (+ 1.0 (/ -1.0 (* x 9.0))) (/ y (* (sqrt x) 3.0))) -5e+17)
   (/ -0.1111111111111111 x)
   1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 + (-1.0 / (x * 9.0))) - (y / (sqrt(x) * 3.0))) <= -5e+17) {
		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 / (sqrt(x) * 3.0d0))) <= (-5d+17)) 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 / (Math.sqrt(x) * 3.0))) <= -5e+17) {
		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 / (math.sqrt(x) * 3.0))) <= -5e+17:
		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(sqrt(x) * 3.0))) <= -5e+17)
		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 / (sqrt(x) * 3.0))) <= -5e+17)
		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[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+17], 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)))) < -5e17

   1. Initial program 99.5%

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

   3. Taylor expanded in 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 61.2

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

   5. Simplified 61.2%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 61.8

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

   8. Simplified 61.8%

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

   if -5e17 < (-.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 61.0

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

   5. Simplified 61.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 60.8%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot 3} \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+55}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+67}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e+55)
   (- 1.0 (/ y (* (sqrt x) 3.0)))
   (if (<= y 1.15e+67)
     (+ 1.0 (/ -0.1111111111111111 x))
     (fma -0.3333333333333333 (/ y (sqrt x)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+55) {
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	} else if (y <= 1.15e+67) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = fma(-0.3333333333333333, (y / sqrt(x)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e+55)
		tmp = Float64(1.0 - Float64(y / Float64(sqrt(x) * 3.0)));
	elseif (y <= 1.15e+67)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = fma(-0.3333333333333333, Float64(y / sqrt(x)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e+55], N[(1.0 - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+67], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.49999999999999989e55

   1. Initial program 99.4%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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 95.4%

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

   if -9.49999999999999989e55 < y < 1.1499999999999999e67

   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 96.7

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

   5. Simplified 96.7%

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

   if 1.1499999999999999e67 < y

   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. 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. times-frac N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. sub-neg N/A

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

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

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-frac N/A

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

      21. /-lowering-/.f64 N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 97.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+55}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+67}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{\sqrt{x}}, 1\right)\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+66}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.3333333333333333 (/ y (sqrt x)) 1.0)))
   (if (<= y -6.8e+62)
     t_0
     (if (<= y 2.9e+66) (+ 1.0 (/ -0.1111111111111111 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(-0.3333333333333333, (y / sqrt(x)), 1.0);
	double tmp;
	if (y <= -6.8e+62) {
		tmp = t_0;
	} else if (y <= 2.9e+66) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.3333333333333333, Float64(y / sqrt(x)), 1.0)
	tmp = 0.0
	if (y <= -6.8e+62)
		tmp = t_0;
	elseif (y <= 2.9e+66)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -6.8e+62], t$95$0, If[LessEqual[y, 2.9e+66], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.80000000000000028e62 or 2.89999999999999986e66 < y

   1. Initial program 99.5%

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

      1. 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. times-frac N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. sub-neg N/A

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

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

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. distribute-neg-frac N/A

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

      21. /-lowering-/.f64 N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 96.5%

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

   if -6.80000000000000028e62 < y < 2.89999999999999986e66

   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 96.7

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

   5. Simplified 96.7%

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

Alternative 5: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 310:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, -0.3333333333333333 \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 310.0)
   (/ (fma (sqrt x) (* -0.3333333333333333 y) -0.1111111111111111) x)
   (- 1.0 (/ y (* (sqrt x) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 310.0) {
		tmp = fma(sqrt(x), (-0.3333333333333333 * y), -0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / (sqrt(x) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 310.0)
		tmp = Float64(fma(sqrt(x), Float64(-0.3333333333333333 * y), -0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / Float64(sqrt(x) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 310.0], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(-0.3333333333333333 * y), $MachinePrecision] + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 310

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. /-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. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 99.5

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

   5. Simplified 99.5%

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

   if 310 < 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 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 310:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, -0.3333333333333333 \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+138}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.012345679012345678}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e+138)
   (+ 1.0 (/ -0.1111111111111111 x))
   (/ -0.012345679012345678 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e+138) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.012345679012345678 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.5d+138) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (-0.012345679012345678d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e+138) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.012345679012345678 / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e+138:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = -0.012345679012345678 / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e+138)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(-0.012345679012345678 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e+138)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = -0.012345679012345678 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e+138], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(-0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.49999999999999982e138

   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 69.9

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

   5. Simplified 69.9%

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

   if 4.49999999999999982e138 < 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 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 3.8

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

   5. Simplified 3.8%

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

      1. metadata-eval N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

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

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

      6. metadata-eval N/A

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

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

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. frac-times N/A

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

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

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

      14. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{81}}}{x \cdot x} - 1}{\frac{1}{x} \cdot \frac{-1}{9} - 1} \]

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

          \[\leadsto \frac{\frac{\frac{1}{81}}{\color{blue}{x \cdot x}} - 1}{\frac{1}{x} \cdot \frac{-1}{9} - 1} \]

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

          \[\leadsto \frac{\frac{\frac{1}{81}}{x \cdot x} - 1}{\color{blue}{\frac{1}{x} \cdot \frac{-1}{9} - 1}} \]

      17. associate-*l/ N/A

          \[\leadsto \frac{\frac{\frac{1}{81}}{x \cdot x} - 1}{\color{blue}{\frac{1 \cdot \frac{-1}{9}}{x}} - 1} \]

      18. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{81}}{x \cdot x} - 1}{\frac{\color{blue}{\frac{-1}{9}}}{x} - 1} \]

      19. /-lowering-/.f64 25.5

          \[\leadsto \frac{\frac{0.012345679012345678}{x \cdot x} - 1}{\color{blue}{\frac{-0.1111111111111111}{x}} - 1} \]

   7. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{\frac{\frac{0.012345679012345678}{x \cdot x} - 1}{\frac{-0.1111111111111111}{x} - 1}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{81}}{{x}^{2}}}}{\frac{\frac{-1}{9}}{x} - 1} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{81}}{{x}^{2}}}}{\frac{\frac{-1}{9}}{x} - 1} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{1}{81}}{\color{blue}{x \cdot x}}}{\frac{\frac{-1}{9}}{x} - 1} \]

      3. *-lowering-*.f64 26.4

         \[\leadsto \frac{\frac{0.012345679012345678}{\color{blue}{x \cdot x}}}{\frac{-0.1111111111111111}{x} - 1} \]

   10. Simplified 26.4%

       \[\leadsto \frac{\color{blue}{\frac{0.012345679012345678}{x \cdot x}}}{\frac{-0.1111111111111111}{x} - 1} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{\frac{-1}{81}}{{x}^{2}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{\frac{-1}{81}}{{x}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{\frac{-1}{81}}{\color{blue}{x \cdot x}} \]

       3. *-lowering-*.f64 27.1

          \[\leadsto \frac{-0.012345679012345678}{\color{blue}{x \cdot x}} \]

   13. Simplified 27.1%

       \[\leadsto \color{blue}{\frac{-0.012345679012345678}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 62.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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 61.1

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

5. Simplified 61.1%

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

Alternative 8: 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 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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 61.1

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

5. Simplified 61.1%

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

6. Taylor expanded in x around inf

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

8. Simplified 32.8%

   \[\leadsto \color{blue}{1} \]
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 2024205 
(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)))))