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

Percentage Accurate: 99.7% → 99.7%

Time: 9.0s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  (- 1.0 (/ 0.1111111111111111 x))
  (* -0.3333333333333333 (* y (pow x -0.5)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y * pow(x, -0.5)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) + ((-0.3333333333333333d0) * (y * (x ** (-0.5d0))))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y * Math.pow(x, -0.5)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y * math.pow(x, -0.5)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) + Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y * (x ^ -0.5)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. clear-num 99.7%

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

   2. associate-/r/ 99.7%

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

   3. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot y\right) \]

   4. pow-flip 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot y\right) \]

   5. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \left({x}^{\color{blue}{-0.5}} \cdot y\right) \]

5. Applied egg-rr 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \color{blue}{\left({x}^{-0.5} \cdot y\right)} \]

6. Final simplification 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right) \]

Alternative 3: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+104} \lor \neg \left(y \leq 1.8 \cdot 10^{+97}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.15e+104) (not (<= y 1.8e+97)))
   (* -0.3333333333333333 (* y (sqrt (/ 1.0 x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.15e+104) || !(y <= 1.8e+97)) {
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / 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 <= (-2.15d+104)) .or. (.not. (y <= 1.8d+97))) then
        tmp = (-0.3333333333333333d0) * (y * sqrt((1.0d0 / 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 <= -2.15e+104) || !(y <= 1.8e+97)) {
		tmp = -0.3333333333333333 * (y * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.15e+104) or not (y <= 1.8e+97):
		tmp = -0.3333333333333333 * (y * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.15e+104) || !(y <= 1.8e+97))
		tmp = Float64(-0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / 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 <= -2.15e+104) || ~((y <= 1.8e+97)))
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.15e+104], N[Not[LessEqual[y, 1.8e+97]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1500000000000001e104 or 1.79999999999999983e97 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

   3. Taylor expanded in y around inf 95.0%

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

   if -2.1500000000000001e104 < y < 1.79999999999999983e97

   1. Initial program 99.9%

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

      1. associate--l- 99.9%

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

      2. +-commutative 99.9%

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

      3. associate--r+ 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. associate-+r- 99.9%

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

      7. neg-mul-1 99.9%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 95.7%

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

      1. cancel-sign-sub-inv 95.7%

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

      2. metadata-eval 95.7%

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

      3. associate-*r/ 95.8%

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

      4. metadata-eval 95.8%

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

      5. +-commutative 95.8%

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

   6. Simplified 95.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+104} \lor \neg \left(y \leq 1.8 \cdot 10^{+97}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 4: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+104} \lor \neg \left(y \leq 1.5 \cdot 10^{+39}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.2e+104) (not (<= y 1.5e+39)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+104) || !(y <= 1.5e+39)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(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 <= (-4.2d+104)) .or. (.not. (y <= 1.5d+39))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(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 <= -4.2e+104) || !(y <= 1.5e+39)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.2e+104) or not (y <= 1.5e+39):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.2e+104) || !(y <= 1.5e+39))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(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 <= -4.2e+104) || ~((y <= 1.5e+39)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.2e+104], N[Not[LessEqual[y, 1.5e+39]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1999999999999997e104 or 1.5e39 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. distribute-neg-frac 99.5%

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

      13. metadata-eval 99.5%

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

      14. *-commutative 99.5%

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

      15. associate-/r* 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 96.4%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. inv-pow 96.4%

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

      2. sqrt-pow1 96.5%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot y\right) \]

      3. metadata-eval 96.5%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left({x}^{\color{blue}{-0.5}} \cdot y\right) \]

      4. expm1-log1p-u 95.3%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      5. expm1-udef 53.6%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]

   6. Applied egg-rr 53.6%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]
   7. Step-by-step derivation

      1. expm1-def 95.3%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      2. expm1-log1p 96.5%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]

   8. Simplified 96.5%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 48.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)\right)} \]

      2. expm1-udef 48.1%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)} - 1\right)} \]

      3. associate-*r* 48.1%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right) \cdot y}\right)} - 1\right) \]

      4. metadata-eval 48.1%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \cdot y\right)} - 1\right) \]

      5. sqrt-pow1 48.1%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \color{blue}{\sqrt{{x}^{-1}}}\right) \cdot y\right)} - 1\right) \]

      6. inv-pow 48.1%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right) \cdot y\right)} - 1\right) \]

      7. sqrt-div 48.1%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right) \cdot y\right)} - 1\right) \]

      8. metadata-eval 48.1%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right) \cdot y\right)} - 1\right) \]

      9. div-inv 48.1%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot y\right)} - 1\right) \]

   10. Applied egg-rr 48.1%

       \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 48.1%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\right)\right)} \]

       2. expm1-log1p 96.4%

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

       3. associate-*l/ 96.4%

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

       4. associate-*r/ 96.4%

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

   12. Simplified 96.4%

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

   if -4.1999999999999997e104 < y < 1.5e39

   1. Initial program 99.9%

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

      1. associate--l- 99.9%

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

      2. +-commutative 99.9%

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

      3. associate--r+ 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. associate-+r- 99.9%

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

      7. neg-mul-1 99.9%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 96.6%

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

      1. cancel-sign-sub-inv 96.6%

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

      2. metadata-eval 96.6%

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

      3. associate-*r/ 96.7%

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

      4. metadata-eval 96.7%

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

      5. +-commutative 96.7%

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

   6. Simplified 96.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+104} \lor \neg \left(y \leq 1.5 \cdot 10^{+39}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 5: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \mathbf{if}\;y \leq -1.75 \cdot 10^{+104}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot t_0\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (if (<= y -1.75e+104)
     (* -0.3333333333333333 (* y t_0))
     (if (<= y 1.7e+97)
       (+ 1.0 (/ -0.1111111111111111 x))
       (* y (* -0.3333333333333333 t_0))))))

C

double code(double x, double y) {
	double t_0 = sqrt((1.0 / x));
	double tmp;
	if (y <= -1.75e+104) {
		tmp = -0.3333333333333333 * (y * t_0);
	} else if (y <= 1.7e+97) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y * (-0.3333333333333333 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / x))
    if (y <= (-1.75d+104)) then
        tmp = (-0.3333333333333333d0) * (y * t_0)
    else if (y <= 1.7d+97) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = y * ((-0.3333333333333333d0) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((1.0 / x));
	double tmp;
	if (y <= -1.75e+104) {
		tmp = -0.3333333333333333 * (y * t_0);
	} else if (y <= 1.7e+97) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y * (-0.3333333333333333 * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((1.0 / x))
	tmp = 0
	if y <= -1.75e+104:
		tmp = -0.3333333333333333 * (y * t_0)
	elif y <= 1.7e+97:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = y * (-0.3333333333333333 * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(1.0 / x))
	tmp = 0.0
	if (y <= -1.75e+104)
		tmp = Float64(-0.3333333333333333 * Float64(y * t_0));
	elseif (y <= 1.7e+97)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((1.0 / x));
	tmp = 0.0;
	if (y <= -1.75e+104)
		tmp = -0.3333333333333333 * (y * t_0);
	elseif (y <= 1.7e+97)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = y * (-0.3333333333333333 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.75e+104], N[(-0.3333333333333333 * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+97], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7500000000000001e104

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.5%

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

   3. Taylor expanded in y around inf 94.2%

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

   if -1.7500000000000001e104 < y < 1.70000000000000005e97

   1. Initial program 99.9%

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

      1. associate--l- 99.9%

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

      2. +-commutative 99.9%

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

      3. associate--r+ 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. associate-+r- 99.9%

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

      7. neg-mul-1 99.9%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 95.7%

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

      1. cancel-sign-sub-inv 95.7%

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

      2. metadata-eval 95.7%

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

      3. associate-*r/ 95.8%

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

      4. metadata-eval 95.8%

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

      5. +-commutative 95.8%

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

   6. Simplified 95.8%

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

   if 1.70000000000000005e97 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

   3. Taylor expanded in y around inf 95.7%

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

      1. associate-*r* 95.8%

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

   5. Simplified 95.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+104}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]

Alternative 6: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e+104)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 1.12e+39)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+104) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 1.12e+39) {
		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 <= (-1.7d+104)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= 1.12d+39) 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 <= -1.7e+104) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 1.12e+39) {
		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 <= -1.7e+104:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 1.12e+39:
		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 <= -1.7e+104)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 1.12e+39)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e+104)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 1.12e+39)
		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, -1.7e+104], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e+39], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999998e104

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. associate--r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.3%

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

      11. metadata-eval 99.3%

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

      12. distribute-neg-frac 99.3%

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

      13. metadata-eval 99.3%

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

      14. *-commutative 99.3%

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

      15. associate-/r* 99.3%

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

      16. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around inf 98.4%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. inv-pow 98.4%

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

      2. sqrt-pow1 98.6%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot y\right) \]

      3. metadata-eval 98.6%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left({x}^{\color{blue}{-0.5}} \cdot y\right) \]

      4. expm1-log1p-u 97.3%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      5. expm1-udef 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]

   6. Applied egg-rr 51.7%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]
   7. Step-by-step derivation

      1. expm1-def 97.3%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      2. expm1-log1p 98.6%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]

   8. Simplified 98.6%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]
   9. Step-by-step derivation

      1. associate-*r* 98.4%

         \[\leadsto 1 + \color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right) \cdot y} \]

      2. metadata-eval 98.4%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \cdot y \]

      3. sqrt-pow1 98.3%

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

      4. inv-pow 98.3%

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

      5. sqrt-div 98.3%

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

      6. metadata-eval 98.3%

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

      7. div-inv 98.3%

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

      8. associate-/r/ 98.4%

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

   10. Applied egg-rr 98.4%

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

   if -1.6999999999999998e104 < y < 1.12e39

   1. Initial program 99.9%

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

      1. associate--l- 99.9%

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

      2. +-commutative 99.9%

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

      3. associate--r+ 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. associate-+r- 99.9%

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

      7. neg-mul-1 99.9%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 96.6%

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

      1. cancel-sign-sub-inv 96.6%

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

      2. metadata-eval 96.6%

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

      3. associate-*r/ 96.7%

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

      4. metadata-eval 96.7%

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

      5. +-commutative 96.7%

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

   6. Simplified 96.7%

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

   if 1.12e39 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.6%

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

      9. fma-neg 99.6%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. distribute-neg-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 94.8%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. inv-pow 94.8%

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

      2. sqrt-pow1 94.8%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot y\right) \]

      3. metadata-eval 94.8%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left({x}^{\color{blue}{-0.5}} \cdot y\right) \]

      4. expm1-log1p-u 93.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      5. expm1-udef 55.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]

   6. Applied egg-rr 55.0%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]
   7. Step-by-step derivation

      1. expm1-def 93.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      2. expm1-log1p 94.8%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]

   8. Simplified 94.8%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 13.7%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)\right)} \]

      2. expm1-udef 13.7%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)} - 1\right)} \]

      3. associate-*r* 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right) \cdot y}\right)} - 1\right) \]

      4. metadata-eval 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \cdot y\right)} - 1\right) \]

      5. sqrt-pow1 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \color{blue}{\sqrt{{x}^{-1}}}\right) \cdot y\right)} - 1\right) \]

      6. inv-pow 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right) \cdot y\right)} - 1\right) \]

      7. sqrt-div 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right) \cdot y\right)} - 1\right) \]

      8. metadata-eval 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right) \cdot y\right)} - 1\right) \]

      9. div-inv 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot y\right)} - 1\right) \]

   10. Applied egg-rr 13.7%

       \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 13.7%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\right)\right)} \]

       2. expm1-log1p 94.9%

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

       3. associate-*l/ 95.0%

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

       4. associate-*r/ 94.9%

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

   12. Simplified 94.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 7: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e+104)
   (+ 1.0 (/ y (* (sqrt x) -3.0)))
   (if (<= y 1.3e+39)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+104) {
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	} else if (y <= 1.3e+39) {
		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 <= (-1.7d+104)) then
        tmp = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    else if (y <= 1.3d+39) 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 <= -1.7e+104) {
		tmp = 1.0 + (y / (Math.sqrt(x) * -3.0));
	} else if (y <= 1.3e+39) {
		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 <= -1.7e+104:
		tmp = 1.0 + (y / (math.sqrt(x) * -3.0))
	elif y <= 1.3e+39:
		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 <= -1.7e+104)
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	elseif (y <= 1.3e+39)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e+104)
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	elseif (y <= 1.3e+39)
		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, -1.7e+104], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+39], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999998e104

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. associate--r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.3%

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

      11. metadata-eval 99.3%

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

      12. distribute-neg-frac 99.3%

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

      13. metadata-eval 99.3%

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

      14. *-commutative 99.3%

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

      15. associate-/r* 99.3%

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

      16. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around inf 98.4%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. inv-pow 98.4%

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

      2. sqrt-pow1 98.6%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot y\right) \]

      3. metadata-eval 98.6%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left({x}^{\color{blue}{-0.5}} \cdot y\right) \]

      4. expm1-log1p-u 97.3%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      5. expm1-udef 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]

   6. Applied egg-rr 51.7%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]
   7. Step-by-step derivation

      1. expm1-def 97.3%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      2. expm1-log1p 98.6%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]

   8. Simplified 98.6%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 91.9%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)\right)} \]

      2. expm1-udef 91.9%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)} - 1\right)} \]

      3. associate-*r* 91.9%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right) \cdot y}\right)} - 1\right) \]

      4. metadata-eval 91.9%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \cdot y\right)} - 1\right) \]

      5. sqrt-pow1 91.9%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \color{blue}{\sqrt{{x}^{-1}}}\right) \cdot y\right)} - 1\right) \]

      6. inv-pow 91.9%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right) \cdot y\right)} - 1\right) \]

      7. sqrt-div 91.9%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right) \cdot y\right)} - 1\right) \]

      8. metadata-eval 91.9%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right) \cdot y\right)} - 1\right) \]

      9. div-inv 91.9%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot y\right)} - 1\right) \]

   10. Applied egg-rr 91.9%

       \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 91.9%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\right)\right)} \]

       2. expm1-log1p 98.3%

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

       3. associate-*l/ 98.3%

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

       4. associate-*r/ 98.3%

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

   12. Simplified 98.3%

       \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
   13. Step-by-step derivation

       1. associate-*r/ 98.3%

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

       2. associate-*l/ 98.3%

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

       3. clear-num 98.1%

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

       4. div-inv 98.4%

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

       5. metadata-eval 98.4%

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

       6. metadata-eval 98.4%

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

       7. distribute-rgt-neg-in 98.4%

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

       8. *-commutative 98.4%

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

       9. associate-*l/ 98.4%

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

       10. *-un-lft-identity 98.4%

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

       11. *-commutative 98.4%

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

       12. distribute-rgt-neg-in 98.4%

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

       13. metadata-eval 98.4%

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

   14. Applied egg-rr 98.4%

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

   if -1.6999999999999998e104 < y < 1.3e39

   1. Initial program 99.9%

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

      1. associate--l- 99.9%

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

      2. +-commutative 99.9%

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

      3. associate--r+ 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. associate-+r- 99.9%

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

      7. neg-mul-1 99.9%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 96.6%

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

      1. cancel-sign-sub-inv 96.6%

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

      2. metadata-eval 96.6%

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

      3. associate-*r/ 96.7%

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

      4. metadata-eval 96.7%

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

      5. +-commutative 96.7%

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

   6. Simplified 96.7%

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

   if 1.3e39 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.6%

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

      9. fma-neg 99.6%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. distribute-neg-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 94.8%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. inv-pow 94.8%

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

      2. sqrt-pow1 94.8%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot y\right) \]

      3. metadata-eval 94.8%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left({x}^{\color{blue}{-0.5}} \cdot y\right) \]

      4. expm1-log1p-u 93.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      5. expm1-udef 55.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]

   6. Applied egg-rr 55.0%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]
   7. Step-by-step derivation

      1. expm1-def 93.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      2. expm1-log1p 94.8%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]

   8. Simplified 94.8%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 13.7%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)\right)} \]

      2. expm1-udef 13.7%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)} - 1\right)} \]

      3. associate-*r* 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right) \cdot y}\right)} - 1\right) \]

      4. metadata-eval 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \cdot y\right)} - 1\right) \]

      5. sqrt-pow1 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \color{blue}{\sqrt{{x}^{-1}}}\right) \cdot y\right)} - 1\right) \]

      6. inv-pow 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right) \cdot y\right)} - 1\right) \]

      7. sqrt-div 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right) \cdot y\right)} - 1\right) \]

      8. metadata-eval 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right) \cdot y\right)} - 1\right) \]

      9. div-inv 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot y\right)} - 1\right) \]

   10. Applied egg-rr 13.7%

       \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 13.7%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\right)\right)} \]

       2. expm1-log1p 94.9%

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

       3. associate-*l/ 95.0%

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

       4. associate-*r/ 94.9%

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

   12. Simplified 94.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 8: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e+104)
   (+ 1.0 (* -0.3333333333333333 (* y (pow x -0.5))))
   (if (<= y 1.45e+39)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+104) {
		tmp = 1.0 + (-0.3333333333333333 * (y * pow(x, -0.5)));
	} else if (y <= 1.45e+39) {
		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 <= (-1.7d+104)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y * (x ** (-0.5d0))))
    else if (y <= 1.45d+39) 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 <= -1.7e+104) {
		tmp = 1.0 + (-0.3333333333333333 * (y * Math.pow(x, -0.5)));
	} else if (y <= 1.45e+39) {
		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 <= -1.7e+104:
		tmp = 1.0 + (-0.3333333333333333 * (y * math.pow(x, -0.5)))
	elif y <= 1.45e+39:
		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 <= -1.7e+104)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5))));
	elseif (y <= 1.45e+39)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e+104)
		tmp = 1.0 + (-0.3333333333333333 * (y * (x ^ -0.5)));
	elseif (y <= 1.45e+39)
		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, -1.7e+104], N[(1.0 + N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e+39], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999998e104

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. associate--r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.3%

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

      11. metadata-eval 99.3%

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

      12. distribute-neg-frac 99.3%

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

      13. metadata-eval 99.3%

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

      14. *-commutative 99.3%

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

      15. associate-/r* 99.3%

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

      16. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around inf 98.4%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. inv-pow 98.4%

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

      2. sqrt-pow1 98.6%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot y\right) \]

      3. metadata-eval 98.6%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left({x}^{\color{blue}{-0.5}} \cdot y\right) \]

      4. expm1-log1p-u 97.3%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      5. expm1-udef 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]

   6. Applied egg-rr 51.7%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]
   7. Step-by-step derivation

      1. expm1-def 97.3%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      2. expm1-log1p 98.6%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]

   8. Simplified 98.6%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]

   if -1.6999999999999998e104 < y < 1.45000000000000015e39

   1. Initial program 99.9%

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

      1. associate--l- 99.9%

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

      2. +-commutative 99.9%

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

      3. associate--r+ 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. associate-+r- 99.9%

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

      7. neg-mul-1 99.9%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 96.6%

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

      1. cancel-sign-sub-inv 96.6%

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

      2. metadata-eval 96.6%

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

      3. associate-*r/ 96.7%

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

      4. metadata-eval 96.7%

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

      5. +-commutative 96.7%

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

   6. Simplified 96.7%

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

   if 1.45000000000000015e39 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.6%

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

      9. fma-neg 99.6%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. distribute-neg-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 94.8%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. inv-pow 94.8%

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

      2. sqrt-pow1 94.8%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot y\right) \]

      3. metadata-eval 94.8%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left({x}^{\color{blue}{-0.5}} \cdot y\right) \]

      4. expm1-log1p-u 93.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      5. expm1-udef 55.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]

   6. Applied egg-rr 55.0%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \]
   7. Step-by-step derivation

      1. expm1-def 93.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \]

      2. expm1-log1p 94.8%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]

   8. Simplified 94.8%

      \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 13.7%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)\right)} \]

      2. expm1-udef 13.7%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\right)} - 1\right)} \]

      3. associate-*r* 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right) \cdot y}\right)} - 1\right) \]

      4. metadata-eval 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot {x}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \cdot y\right)} - 1\right) \]

      5. sqrt-pow1 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \color{blue}{\sqrt{{x}^{-1}}}\right) \cdot y\right)} - 1\right) \]

      6. inv-pow 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right) \cdot y\right)} - 1\right) \]

      7. sqrt-div 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right) \cdot y\right)} - 1\right) \]

      8. metadata-eval 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\left(-0.3333333333333333 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right) \cdot y\right)} - 1\right) \]

      9. div-inv 13.7%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot y\right)} - 1\right) \]

   10. Applied egg-rr 13.7%

       \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 13.7%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333}{\sqrt{x}} \cdot y\right)\right)} \]

       2. expm1-log1p 94.9%

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

       3. associate-*l/ 95.0%

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

       4. associate-*r/ 94.9%

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

   12. Simplified 94.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 9: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) + ((-0.3333333333333333d0) * (y / sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / math.sqrt(x)))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / sqrt(x)));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) + ((-0.3333333333333333d0) / (sqrt(x) / y))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 / (Math.sqrt(x) / y));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 / (math.sqrt(x) / y))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 / (sqrt(x) / y));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. clear-num 99.7%

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

   2. un-div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}} \]

Alternative 11: 64.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+168}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{-0.012345679012345678}{x \cdot \left(-x\right)}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.6e+168)
   (+ 1.0 (/ -0.1111111111111111 x))
   (/
    (- 1.0 (/ -0.012345679012345678 (* x (- x))))
    (- 1.0 (/ -0.1111111111111111 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.6e+168) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - (-0.012345679012345678 / (x * -x))) / (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 <= 4.6d+168) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 - ((-0.012345679012345678d0) / (x * -x))) / (1.0d0 - ((-0.1111111111111111d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.6e+168) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - (-0.012345679012345678 / (x * -x))) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.6e+168:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 - (-0.012345679012345678 / (x * -x))) / (1.0 - (-0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.6e+168)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 - Float64(-0.012345679012345678 / Float64(x * Float64(-x)))) / Float64(1.0 - Float64(-0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.6e+168)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 - (-0.012345679012345678 / (x * -x))) / (1.0 - (-0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.6e+168], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(-0.012345679012345678 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5999999999999999e168

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. distribute-neg-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 75.2%

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

      1. cancel-sign-sub-inv 75.2%

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

      2. metadata-eval 75.2%

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

      3. associate-*r/ 75.3%

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

      4. metadata-eval 75.3%

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

      5. +-commutative 75.3%

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

   6. Simplified 75.3%

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

   if 4.5999999999999999e168 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.4%

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

      9. fma-neg 99.4%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. distribute-neg-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 4.2%

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

      1. cancel-sign-sub-inv 4.2%

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

      2. metadata-eval 4.2%

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

      3. associate-*r/ 4.2%

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

      4. metadata-eval 4.2%

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

      5. +-commutative 4.2%

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

   6. Simplified 4.2%

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

      1. div-inv 4.2%

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

      2. fma-def 4.2%

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

   8. Applied egg-rr 4.2%

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

      1. fma-udef 4.2%

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

      2. metadata-eval 4.2%

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

      3. distribute-lft-neg-in 4.2%

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

      4. div-inv 4.2%

         \[\leadsto \left(-\color{blue}{\frac{0.1111111111111111}{x}}\right) + 1 \]

      5. +-commutative 4.2%

         \[\leadsto \color{blue}{1 + \left(-\frac{0.1111111111111111}{x}\right)} \]

      6. flip-+ 39.1%

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

      7. metadata-eval 39.1%

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

      8. div-inv 39.1%

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

      9. distribute-lft-neg-in 39.1%

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

      10. metadata-eval 39.1%

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

      11. div-inv 39.1%

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

      12. distribute-lft-neg-in 39.1%

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

      13. metadata-eval 39.1%

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

      14. un-div-inv 39.1%

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

      15. un-div-inv 39.1%

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

      16. distribute-neg-frac 39.1%

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

      17. metadata-eval 39.1%

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

   10. Applied egg-rr 39.1%

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

       1. frac-2neg 39.1%

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

       2. frac-times 39.1%

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

       3. metadata-eval 39.1%

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

       4. metadata-eval 39.1%

          \[\leadsto \frac{1 - \frac{\color{blue}{-0.012345679012345678}}{\left(-x\right) \cdot x}}{1 - \frac{-0.1111111111111111}{x}} \]

   12. Applied egg-rr 39.1%

       \[\leadsto \frac{1 - \color{blue}{\frac{-0.012345679012345678}{\left(-x\right) \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+168}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{-0.012345679012345678}{x \cdot \left(-x\right)}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]

Alternative 12: 64.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+124}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7e+124)
   (+ 1.0 (/ -0.1111111111111111 x))
   (/
    (- 1.0 (/ (/ 0.012345679012345678 x) x))
    (- 1.0 (/ -0.1111111111111111 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e+124) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (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 <= 7d+124) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 - ((0.012345679012345678d0 / x) / x)) / (1.0d0 - ((-0.1111111111111111d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7e+124) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e+124:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e+124)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.012345679012345678 / x) / x)) / Float64(1.0 - Float64(-0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e+124)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 - ((0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e+124], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.0000000000000002e124

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. distribute-neg-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 77.5%

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

      1. cancel-sign-sub-inv 77.5%

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

      2. metadata-eval 77.5%

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

      3. associate-*r/ 77.6%

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

      4. metadata-eval 77.6%

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

      5. +-commutative 77.6%

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

   6. Simplified 77.6%

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

   if 7.0000000000000002e124 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

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

      12. distribute-neg-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 4.1%

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

      1. cancel-sign-sub-inv 4.1%

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

      2. metadata-eval 4.1%

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

      3. associate-*r/ 4.1%

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

      4. metadata-eval 4.1%

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

      5. +-commutative 4.1%

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

   6. Simplified 4.1%

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

      1. div-inv 4.1%

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

      2. fma-def 4.1%

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

   8. Applied egg-rr 4.1%

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

      1. fma-udef 4.1%

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

      2. metadata-eval 4.1%

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

      3. distribute-lft-neg-in 4.1%

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

      4. div-inv 4.1%

         \[\leadsto \left(-\color{blue}{\frac{0.1111111111111111}{x}}\right) + 1 \]

      5. +-commutative 4.1%

         \[\leadsto \color{blue}{1 + \left(-\frac{0.1111111111111111}{x}\right)} \]

      6. flip-+ 32.2%

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

      7. metadata-eval 32.2%

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

      8. div-inv 32.2%

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

      9. distribute-lft-neg-in 32.2%

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

      10. metadata-eval 32.2%

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

      11. div-inv 32.2%

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

      12. distribute-lft-neg-in 32.2%

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

      13. metadata-eval 32.2%

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

      14. un-div-inv 32.2%

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

      15. un-div-inv 32.2%

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

      16. distribute-neg-frac 32.2%

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

      17. metadata-eval 32.2%

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

   10. Applied egg-rr 32.2%

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

       1. frac-times 32.2%

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

       2. associate-/r* 32.2%

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

       3. metadata-eval 32.2%

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

   12. Applied egg-rr 32.2%

       \[\leadsto \frac{1 - \color{blue}{\frac{\frac{0.012345679012345678}{x}}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+124}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]

Alternative 13: 61.3% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2300:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2300.0) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2300.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2300.0d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2300.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2300.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2300.0)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2300.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2300.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2300

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \left(\frac{1}{\color{blue}{{x}^{0.5}}} \cdot y\right) \]

      4. pow-flip 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \left(\color{blue}{{x}^{\left(-0.5\right)}} \cdot y\right) \]

      5. metadata-eval 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \left({x}^{\color{blue}{-0.5}} \cdot y\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \color{blue}{\left({x}^{-0.5} \cdot y\right)} \]

   6. Taylor expanded in x around 0 65.4%

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

   if 2300 < x

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 68.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2300:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 62.3% accurate, 22.6× 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 99.8%

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

   1. associate--l- 99.8%

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

   2. +-commutative 99.8%

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

   3. associate--r+ 99.8%

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

   4. sub-neg 99.8%

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

   5. distribute-frac-neg 99.8%

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

   6. associate-+r- 99.8%

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

   7. neg-mul-1 99.8%

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

   8. associate-*l/ 99.7%

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

   9. fma-neg 99.7%

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

   10. associate-/r* 99.7%

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

   11. metadata-eval 99.7%

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

   12. distribute-neg-frac 99.7%

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

   13. metadata-eval 99.7%

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

   14. *-commutative 99.7%

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

   15. associate-/r* 99.7%

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

   16. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in y around 0 67.1%

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

   1. cancel-sign-sub-inv 67.1%

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

   2. metadata-eval 67.1%

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

   3. associate-*r/ 67.2%

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

   4. metadata-eval 67.2%

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

   5. +-commutative 67.2%

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

6. Simplified 67.2%

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

7. Final simplification 67.2%

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

Alternative 15: 31.7% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.8%

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

   1. associate--l- 99.8%

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

   2. +-commutative 99.8%

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

   3. associate--r+ 99.8%

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

   4. sub-neg 99.8%

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

   5. distribute-frac-neg 99.8%

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

   6. associate-+r- 99.8%

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

   7. neg-mul-1 99.8%

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

   8. associate-*l/ 99.7%

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

   9. fma-neg 99.7%

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

   10. associate-/r* 99.7%

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

   11. metadata-eval 99.7%

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

   12. distribute-neg-frac 99.7%

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

   13. metadata-eval 99.7%

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

   14. *-commutative 99.7%

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

   15. associate-/r* 99.7%

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

   16. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around inf 36.8%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 36.8%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× 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 2023318 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))