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

Percentage Accurate: 99.7% → 99.7%

Time: 5.7s

Alternatives: 11

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 64.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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)))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 6.3%

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

   10. Applied rewrites 100.0%

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

   if -inf.0 < (-.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 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.5%

      \[\leadsto \frac{x - 0.1111111111111111}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+25}:\\ \;\;\;\;\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot y, \sqrt{{x}^{-1}}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6e+25)
   (/ (- x (fma (* (sqrt x) y) 0.3333333333333333 0.1111111111111111)) x)
   (fma (* -0.3333333333333333 y) (sqrt (pow x -1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6e+25) {
		tmp = (x - fma((sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x;
	} else {
		tmp = fma((-0.3333333333333333 * y), sqrt(pow(x, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6e+25)
		tmp = Float64(Float64(x - fma(Float64(sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x);
	else
		tmp = fma(Float64(-0.3333333333333333 * y), sqrt((x ^ -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6e+25], N[(N[(x - N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 0.3333333333333333 + 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(-0.3333333333333333 * y), $MachinePrecision] * N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.00000000000000011e25

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 6.00000000000000011e25 < x

   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. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333 \cdot y, \sqrt{\frac{1}{x}}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+25}:\\ \;\;\;\;\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot y, \sqrt{{x}^{-1}}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

   3. 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

      \[\leadsto \frac{-0.1111111111111111 - 0.3333333333333333 \cdot \left(\sqrt{x} \cdot y\right)}{x} \]

   if 0.110000000000000001 < x

   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. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333 \cdot y, \sqrt{\frac{1}{x}}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

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

Alternative 6: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

   3. 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.110000000000000001 < x

   1. Initial program 99.7%

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

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333 \cdot y, \sqrt{\frac{1}{x}}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

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

Alternative 7: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.46 \cdot 10^{+56} \lor \neg \left(y \leq 2.7 \cdot 10^{+49}\right):\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.46e+56) (not (<= y 2.7e+49)))
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (/ (- x 0.1111111111111111) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.46e+56) || !(y <= 2.7e+49)) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else {
		tmp = (x - 0.1111111111111111) / 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 <= (-1.46d+56)) .or. (.not. (y <= 2.7d+49))) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else
        tmp = (x - 0.1111111111111111d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.46e+56) || !(y <= 2.7e+49)) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else {
		tmp = (x - 0.1111111111111111) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.46e+56) or not (y <= 2.7e+49):
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	else:
		tmp = (x - 0.1111111111111111) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.46e+56) || !(y <= 2.7e+49))
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	else
		tmp = Float64(Float64(x - 0.1111111111111111) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.46e+56) || ~((y <= 2.7e+49)))
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	else
		tmp = (x - 0.1111111111111111) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.46e+56], N[Not[LessEqual[y, 2.7e+49]], $MachinePrecision]], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4599999999999999e56 or 2.7000000000000001e49 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 94.1%

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

   if -1.4599999999999999e56 < y < 2.7000000000000001e49

   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 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.5%

      \[\leadsto \frac{x - 0.1111111111111111}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.46 \cdot 10^{+56} \lor \neg \left(y \leq 2.7 \cdot 10^{+49}\right):\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - \mathsf{fma}\left(\sqrt{x} \cdot y, 0.3333333333333333, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e+23)
   (/ (- x (fma (* (sqrt x) y) 0.3333333333333333 0.1111111111111111)) x)
   (- 1.0 (/ (/ y (sqrt x)) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e+23) {
		tmp = (x - fma((sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x;
	} else {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e+23)
		tmp = Float64(Float64(x - fma(Float64(sqrt(x) * y), 0.3333333333333333, 0.1111111111111111)) / x);
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e+23], N[(N[(x - N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] * 0.3333333333333333 + 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999999e23

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 4.9999999999999999e23 < x

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.8%

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

Alternative 9: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

   3. 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.110000000000000001 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.9%

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

Alternative 10: 62.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 91.1

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

5. Applied rewrites 91.1%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 59.4%

   \[\leadsto \frac{x - 0.1111111111111111}{x} \]
9. Add Preprocessing

Alternative 11: 31.7% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 62.9

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

5. Applied rewrites 62.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 32.0%

   \[\leadsto \frac{-0.1111111111111111}{x} \]
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 2024320 
(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)))))