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

Percentage Accurate: 99.7% → 99.6%

Time: 6.4s

Alternatives: 10

Speedup: 0.3×

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{x} \cdot y, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.8e+15)
   (- 1.0 (/ (fma 0.3333333333333333 (* (sqrt x) y) 0.1111111111111111) x))
   (- 1.0 (/ (/ y (sqrt x)) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.8e+15) {
		tmp = 1.0 - (fma(0.3333333333333333, (sqrt(x) * 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 <= 1.8e+15)
		tmp = Float64(1.0 - Float64(fma(0.3333333333333333, Float64(sqrt(x) * y), 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, 1.8e+15], N[(1.0 - N[(N[(0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $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 < 1.8e15

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

   if 1.8e15 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{x} \cdot y, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

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(Float64(y / 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[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) - (y / (sqrt(x) * 3.0d0))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / (Math.sqrt(x) * 3.0));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) - (y / (math.sqrt(x) * 3.0))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) - (y / (sqrt(x) * 3.0));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 3.0), $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. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 4: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+37} \lor \neg \left(y \leq 5.8 \cdot 10^{+71}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{9 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.2e+37) (not (<= y 5.8e+71)))
   (- 1.0 (/ y (sqrt (* 9.0 x))))
   (- 1.0 (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e+37) || !(y <= 5.8e+71)) {
		tmp = 1.0 - (y / sqrt((9.0 * x)));
	} else {
		tmp = 1.0 - (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 <= (-1.2d+37)) .or. (.not. (y <= 5.8d+71))) then
        tmp = 1.0d0 - (y / sqrt((9.0d0 * x)))
    else
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e+37) || !(y <= 5.8e+71)) {
		tmp = 1.0 - (y / Math.sqrt((9.0 * x)));
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.2e+37) or not (y <= 5.8e+71):
		tmp = 1.0 - (y / math.sqrt((9.0 * x)))
	else:
		tmp = 1.0 - (0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.2e+37) || !(y <= 5.8e+71))
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(9.0 * x))));
	else
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.2e+37) || ~((y <= 5.8e+71)))
		tmp = 1.0 - (y / sqrt((9.0 * x)));
	else
		tmp = 1.0 - (0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.2e+37], N[Not[LessEqual[y, 5.8e+71]], $MachinePrecision]], N[(1.0 - N[(y / N[Sqrt[N[(9.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e37 or 5.80000000000000014e71 < y

   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. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 96.5

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

         \[\leadsto 1 - \frac{y}{{\left(\sqrt{x} \cdot 3\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. sqrt-pow1 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{{\left(\sqrt{x} \cdot 3\right)}^{2}}}} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-sqrt.f64 96.5

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

   if -1.2e37 < y < 5.80000000000000014e71

   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 97.8%

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

   10. Applied rewrites 97.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+37} \lor \neg \left(y \leq 5.8 \cdot 10^{+71}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{9 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+37}:\\ \;\;\;\;1 - \frac{y}{\sqrt{9 \cdot x}}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+71}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333 \cdot y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+37)
   (- 1.0 (/ y (sqrt (* 9.0 x))))
   (if (<= y 5.8e+71)
     (- 1.0 (/ 0.1111111111111111 x))
     (- 1.0 (/ (* 0.3333333333333333 y) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+37) {
		tmp = 1.0 - (y / sqrt((9.0 * x)));
	} else if (y <= 5.8e+71) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = 1.0 - ((0.3333333333333333 * y) / sqrt(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 <= (-1.2d+37)) then
        tmp = 1.0d0 - (y / sqrt((9.0d0 * x)))
    else if (y <= 5.8d+71) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = 1.0d0 - ((0.3333333333333333d0 * y) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+37) {
		tmp = 1.0 - (y / Math.sqrt((9.0 * x)));
	} else if (y <= 5.8e+71) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = 1.0 - ((0.3333333333333333 * y) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e+37:
		tmp = 1.0 - (y / math.sqrt((9.0 * x)))
	elif y <= 5.8e+71:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = 1.0 - ((0.3333333333333333 * y) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+37)
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(9.0 * x))));
	elseif (y <= 5.8e+71)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = Float64(1.0 - Float64(Float64(0.3333333333333333 * y) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e+37)
		tmp = 1.0 - (y / sqrt((9.0 * x)));
	elseif (y <= 5.8e+71)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = 1.0 - ((0.3333333333333333 * y) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e+37], N[(1.0 - N[(y / N[Sqrt[N[(9.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e+71], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.3333333333333333 * y), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e37

   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 94.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 94.0

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

         \[\leadsto 1 - \frac{y}{{\left(\sqrt{x} \cdot 3\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. sqrt-pow1 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{{\left(\sqrt{x} \cdot 3\right)}^{2}}}} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-sqrt.f64 94.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 94.0

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

   7. Applied rewrites 94.0%

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

   if -1.2e37 < y < 5.80000000000000014e71

   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 97.8%

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

   10. Applied rewrites 97.8%

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

   if 5.80000000000000014e71 < y

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 99.3

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

   10. Applied rewrites 99.3%

       \[\leadsto 1 - \frac{\color{blue}{0.3333333333333333 \cdot y}}{\sqrt{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+37}:\\ \;\;\;\;1 - \frac{y}{\sqrt{9 \cdot x}}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+71}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333 \cdot y}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+37}:\\ \;\;\;\;1 - \frac{y}{\sqrt{9 \cdot x}}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+71}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+37)
   (- 1.0 (/ y (sqrt (* 9.0 x))))
   (if (<= y 5.8e+71)
     (- 1.0 (/ 0.1111111111111111 x))
     (- 1.0 (/ y (* 3.0 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+37) {
		tmp = 1.0 - (y / sqrt((9.0 * x)));
	} else if (y <= 5.8e+71) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(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 <= (-1.2d+37)) then
        tmp = 1.0d0 - (y / sqrt((9.0d0 * x)))
    else if (y <= 5.8d+71) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+37) {
		tmp = 1.0 - (y / Math.sqrt((9.0 * x)));
	} else if (y <= 5.8e+71) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e+37:
		tmp = 1.0 - (y / math.sqrt((9.0 * x)))
	elif y <= 5.8e+71:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+37)
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(9.0 * x))));
	elseif (y <= 5.8e+71)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e+37)
		tmp = 1.0 - (y / sqrt((9.0 * x)));
	elseif (y <= 5.8e+71)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e+37], N[(1.0 - N[(y / N[Sqrt[N[(9.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e+71], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e37

   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 94.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 94.0

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

         \[\leadsto 1 - \frac{y}{{\left(\sqrt{x} \cdot 3\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. sqrt-pow1 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{{\left(\sqrt{x} \cdot 3\right)}^{2}}}} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-sqrt.f64 94.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 94.0

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

   7. Applied rewrites 94.0%

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

   if -1.2e37 < y < 5.80000000000000014e71

   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 97.8%

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

   10. Applied rewrites 97.8%

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

   if 5.80000000000000014e71 < y

   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. Applied rewrites 99.3%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+37}:\\ \;\;\;\;1 - \frac{y}{\sqrt{9 \cdot x}}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+71}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{x} \cdot y, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{9 \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.8e+15)
   (- 1.0 (/ (fma 0.3333333333333333 (* (sqrt x) y) 0.1111111111111111) x))
   (- 1.0 (/ y (sqrt (* 9.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.8e+15) {
		tmp = 1.0 - (fma(0.3333333333333333, (sqrt(x) * y), 0.1111111111111111) / x);
	} else {
		tmp = 1.0 - (y / sqrt((9.0 * x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.8e+15)
		tmp = Float64(1.0 - Float64(fma(0.3333333333333333, Float64(sqrt(x) * y), 0.1111111111111111) / x));
	else
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(9.0 * x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.8e+15], N[(1.0 - N[(N[(0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(9.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8e15

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

   if 1.8e15 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 99.8

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

         \[\leadsto 1 - \frac{y}{{\left(\sqrt{x} \cdot 3\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. sqrt-pow1 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{{\left(\sqrt{x} \cdot 3\right)}^{2}}}} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-sqrt.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{x} \cdot y, 0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{9 \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{9 \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (fma -0.3333333333333333 (* (sqrt x) y) -0.1111111111111111) x)
   (- 1.0 (/ y (sqrt (* 9.0 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 / sqrt((9.0 * 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 / sqrt(Float64(9.0 * 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[Sqrt[N[(9.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. 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-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 98.4

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

         \[\leadsto 1 - \frac{y}{{\left(\sqrt{x} \cdot 3\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. sqrt-pow1 N/A

         \[\leadsto 1 - \frac{y}{\color{blue}{\sqrt{{\left(\sqrt{x} \cdot 3\right)}^{2}}}} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-sqrt.f64 98.4

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt{x} \cdot y, -0.1111111111111111\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{9 \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.1% 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 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 94.8

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

5. Applied rewrites 94.8%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 61.0%

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

10. Applied rewrites 61.0%

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

11. Final simplification 61.0%

    \[\leadsto 1 - \frac{0.1111111111111111}{x} \]
12. Add Preprocessing

Alternative 10: 31.5% 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 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

   1. 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 64.1

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

7. Applied rewrites 64.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 30.3%

    \[\leadsto \frac{-0.1111111111111111}{x} \]
11. 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 2024339 
(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)))))