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

Percentage Accurate: 99.7% → 99.7%

Time: 10.5s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+38} \lor \neg \left(y \leq 1.12 \cdot 10^{+36}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.45e+38) (not (<= y 1.12e+36)))
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.45e+38) || !(y <= 1.12e+36)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / 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 <= (-2.45d+38)) .or. (.not. (y <= 1.12d+36))) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / 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 <= -2.45e+38) || !(y <= 1.12e+36)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.45e+38) or not (y <= 1.12e+36):
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.45e+38) || !(y <= 1.12e+36))
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / 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 <= -2.45e+38) || ~((y <= 1.12e+36)))
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.45e+38], N[Not[LessEqual[y, 1.12e+36]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 / 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 < -2.45000000000000001e38 or 1.11999999999999999e36 < y

   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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      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. *-commutative 99.5%

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

      9. associate-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 89.3%

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

      1. associate-*r* 89.3%

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

   7. Simplified 89.3%

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

      1. sqrt-div 89.2%

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

      2. metadata-eval 89.2%

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

      3. div-inv 89.3%

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

      4. expm1-log1p-u 46.2%

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

      5. expm1-udef 8.4%

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

   9. Applied egg-rr 8.4%

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

       1. expm1-def 46.2%

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

       2. expm1-log1p 89.3%

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

   11. Simplified 89.3%

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

   if -2.45000000000000001e38 < y < 1.11999999999999999e36

   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%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      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. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+38} \lor \neg \left(y \leq 1.12 \cdot 10^{+36}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+36} \lor \neg \left(y \leq 9.2 \cdot 10^{+35}\right):\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.2e+36) (not (<= y 9.2e+35)))
   (+ 1.0 (/ (/ y -3.0) (sqrt x)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.2e+36) || !(y <= 9.2e+35)) {
		tmp = 1.0 + ((y / -3.0) / 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 <= (-8.2d+36)) .or. (.not. (y <= 9.2d+35))) then
        tmp = 1.0d0 + ((y / (-3.0d0)) / 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 <= -8.2e+36) || !(y <= 9.2e+35)) {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.2e+36) or not (y <= 9.2e+35):
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.2e+36) || !(y <= 9.2e+35))
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / 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 <= -8.2e+36) || ~((y <= 9.2e+35)))
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.2e+36], N[Not[LessEqual[y, 9.2e+35]], $MachinePrecision]], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.20000000000000026e36 or 9.1999999999999993e35 < y

   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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      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. *-commutative 99.5%

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

      9. associate-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 89.3%

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

      1. associate-*r* 89.3%

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

   7. Simplified 89.3%

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

      1. sqrt-div 89.2%

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

      2. metadata-eval 89.2%

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

      3. div-inv 89.3%

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

      4. expm1-log1p-u 46.2%

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

      5. expm1-udef 8.4%

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

   9. Applied egg-rr 8.4%

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

       1. expm1-def 46.2%

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

       2. expm1-log1p 89.3%

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

   11. Simplified 89.3%

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

       1. associate-*l/ 89.4%

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

       2. *-un-lft-identity 89.4%

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

       3. times-frac 89.4%

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

       4. metadata-eval 89.4%

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

       5. metadata-eval 89.4%

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

       6. times-frac 89.4%

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

       7. *-un-lft-identity 89.4%

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

       8. associate-/r* 89.4%

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

   13. Applied egg-rr 89.4%

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

   if -8.20000000000000026e36 < y < 9.1999999999999993e35

   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%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      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. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+36} \lor \neg \left(y \leq 9.2 \cdot 10^{+35}\right):\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+38}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+36}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e+38)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 1.2e+36)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+38) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 1.2e+36) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / 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 <= (-2.2d+38)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= 1.2d+36) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+38) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 1.2e+36) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e+38:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 1.2e+36:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e+38)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 1.2e+36)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e+38)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 1.2e+36)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e+38], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+36], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.20000000000000006e38

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r- 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r- 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r/ 99.3%

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

      10. fma-neg 99.3%

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

      11. associate-/r* 99.2%

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

      12. metadata-eval 99.2%

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

      13. *-commutative 99.2%

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

      14. associate-/r* 99.2%

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

      15. distribute-neg-frac 99.2%

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

      16. metadata-eval 99.2%

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

      17. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 86.4%

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

      1. associate-*r* 86.4%

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

   7. Simplified 86.4%

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

      1. sqrt-div 86.3%

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

      2. metadata-eval 86.3%

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

      3. div-inv 86.3%

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

      4. expm1-log1p-u 53.8%

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

      5. expm1-udef 10.6%

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

   9. Applied egg-rr 10.6%

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

       1. expm1-def 53.8%

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

       2. expm1-log1p 86.3%

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

   11. Simplified 86.3%

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

       1. associate-*l/ 86.4%

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

       2. associate-/l* 86.4%

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

   13. Applied egg-rr 86.4%

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

   if -2.20000000000000006e38 < y < 1.19999999999999996e36

   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%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      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. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.3%

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

   if 1.19999999999999996e36 < y

   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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      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. *-commutative 99.5%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 92.5%

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

      1. associate-*r* 92.4%

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

   7. Simplified 92.4%

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

      1. sqrt-div 92.4%

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

      2. metadata-eval 92.4%

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

      3. div-inv 92.6%

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

      4. expm1-log1p-u 38.0%

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

      5. expm1-udef 6.0%

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

   9. Applied egg-rr 6.0%

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

       1. expm1-def 38.0%

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

       2. expm1-log1p 92.6%

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

   11. Simplified 92.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+38}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+36}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+38}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+35}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.85e+38)
   (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x)))
   (if (<= y 5.3e+35)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.85e+38) {
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	} else if (y <= 5.3e+35) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / 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 <= (-1.85d+38)) then
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / sqrt(x))
    else if (y <= 5.3d+35) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.85e+38) {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	} else if (y <= 5.3e+35) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.85e+38:
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	elif y <= 5.3e+35:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.85e+38)
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	elseif (y <= 5.3e+35)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.85e+38)
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	elseif (y <= 5.3e+35)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.85e+38], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.3e+35], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8500000000000001e38

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r- 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r- 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r/ 99.3%

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

      10. fma-neg 99.3%

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

      11. associate-/r* 99.2%

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

      12. metadata-eval 99.2%

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

      13. *-commutative 99.2%

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

      14. associate-/r* 99.2%

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

      15. distribute-neg-frac 99.2%

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

      16. metadata-eval 99.2%

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

      17. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 86.4%

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

      1. associate-*r* 86.4%

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

   7. Simplified 86.4%

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

      1. sqrt-div 86.3%

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

      2. metadata-eval 86.3%

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

      3. div-inv 86.3%

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

      4. expm1-log1p-u 53.8%

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

      5. expm1-udef 10.6%

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

   9. Applied egg-rr 10.6%

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

       1. expm1-def 53.8%

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

       2. expm1-log1p 86.3%

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

   11. Simplified 86.3%

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

       1. associate-*l/ 86.4%

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

   13. Applied egg-rr 86.4%

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

   if -1.8500000000000001e38 < y < 5.30000000000000009e35

   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%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      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. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.3%

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

   if 5.30000000000000009e35 < y

   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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      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. *-commutative 99.5%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 92.5%

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

      1. associate-*r* 92.4%

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

   7. Simplified 92.4%

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

      1. sqrt-div 92.4%

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

      2. metadata-eval 92.4%

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

      3. div-inv 92.6%

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

      4. expm1-log1p-u 38.0%

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

      5. expm1-udef 6.0%

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

   9. Applied egg-rr 6.0%

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

       1. expm1-def 38.0%

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

       2. expm1-log1p 92.6%

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

   11. Simplified 92.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+38}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+35}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+36}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+36}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.2e+36)
   (+ 1.0 (/ (/ y (sqrt x)) -3.0))
   (if (<= y 1.28e+36)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (/ (/ y -3.0) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.2e+36) {
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	} else if (y <= 1.28e+36) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9.2d+36)) then
        tmp = 1.0d0 + ((y / sqrt(x)) / (-3.0d0))
    else if (y <= 1.28d+36) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.2e+36) {
		tmp = 1.0 + ((y / Math.sqrt(x)) / -3.0);
	} else if (y <= 1.28e+36) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.2e+36:
		tmp = 1.0 + ((y / math.sqrt(x)) / -3.0)
	elif y <= 1.28e+36:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.2e+36)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) / -3.0));
	elseif (y <= 1.28e+36)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.2e+36)
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	elseif (y <= 1.28e+36)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.2e+36], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.28e+36], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.19999999999999986e36

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r- 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r- 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r/ 99.3%

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

      10. fma-neg 99.3%

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

      11. associate-/r* 99.2%

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

      12. metadata-eval 99.2%

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

      13. *-commutative 99.2%

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

      14. associate-/r* 99.2%

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

      15. distribute-neg-frac 99.2%

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

      16. metadata-eval 99.2%

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

      17. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 86.4%

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

      1. associate-*r* 86.4%

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

   7. Simplified 86.4%

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

      1. sqrt-div 86.3%

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

      2. metadata-eval 86.3%

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

      3. div-inv 86.3%

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

      4. expm1-log1p-u 53.8%

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

      5. expm1-udef 10.6%

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

   9. Applied egg-rr 10.6%

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

       1. expm1-def 53.8%

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

       2. expm1-log1p 86.3%

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

   11. Simplified 86.3%

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

       1. div-inv 86.3%

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

       2. *-commutative 86.3%

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

       3. metadata-eval 86.3%

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

       4. div-inv 86.5%

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

       5. associate-/r* 86.4%

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

       6. *-commutative 86.4%

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

       7. div-inv 86.5%

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

       8. associate-/r* 86.5%

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

   13. Applied egg-rr 86.5%

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

   if -9.19999999999999986e36 < y < 1.27999999999999993e36

   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%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      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. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.3%

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

   if 1.27999999999999993e36 < y

   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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      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. *-commutative 99.5%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 92.5%

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

      1. associate-*r* 92.4%

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

   7. Simplified 92.4%

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

      1. sqrt-div 92.4%

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

      2. metadata-eval 92.4%

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

      3. div-inv 92.6%

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

      4. expm1-log1p-u 38.0%

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

      5. expm1-udef 6.0%

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

   9. Applied egg-rr 6.0%

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

       1. expm1-def 38.0%

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

       2. expm1-log1p 92.6%

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

   11. Simplified 92.6%

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

       1. associate-*l/ 92.5%

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

       2. *-un-lft-identity 92.5%

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

       3. times-frac 92.6%

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

       4. metadata-eval 92.6%

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

       5. metadata-eval 92.6%

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

       6. times-frac 92.5%

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

       7. *-un-lft-identity 92.5%

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

       8. associate-/r* 92.6%

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

   13. Applied egg-rr 92.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+36}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+36}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+38}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+35}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e+38)
   (+ 1.0 (/ (/ y (sqrt x)) -3.0))
   (if (<= y 9.8e+35)
     (- 1.0 (pow (* x 9.0) -1.0))
     (+ 1.0 (/ (/ y -3.0) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e+38) {
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	} else if (y <= 9.8e+35) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.4d+38)) then
        tmp = 1.0d0 + ((y / sqrt(x)) / (-3.0d0))
    else if (y <= 9.8d+35) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4e+38) {
		tmp = 1.0 + ((y / Math.sqrt(x)) / -3.0);
	} else if (y <= 9.8e+35) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e+38:
		tmp = 1.0 + ((y / math.sqrt(x)) / -3.0)
	elif y <= 9.8e+35:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e+38)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) / -3.0));
	elseif (y <= 9.8e+35)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e+38)
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	elseif (y <= 9.8e+35)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e+38], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.8e+35], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.40000000000000017e38

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r- 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r- 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r/ 99.3%

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

      10. fma-neg 99.3%

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

      11. associate-/r* 99.2%

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

      12. metadata-eval 99.2%

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

      13. *-commutative 99.2%

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

      14. associate-/r* 99.2%

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

      15. distribute-neg-frac 99.2%

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

      16. metadata-eval 99.2%

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

      17. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 86.4%

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

      1. associate-*r* 86.4%

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

   7. Simplified 86.4%

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

      1. sqrt-div 86.3%

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

      2. metadata-eval 86.3%

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

      3. div-inv 86.3%

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

      4. expm1-log1p-u 53.8%

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

      5. expm1-udef 10.6%

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

   9. Applied egg-rr 10.6%

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

       1. expm1-def 53.8%

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

       2. expm1-log1p 86.3%

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

   11. Simplified 86.3%

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

       1. div-inv 86.3%

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

       2. *-commutative 86.3%

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

       3. metadata-eval 86.3%

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

       4. div-inv 86.5%

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

       5. associate-/r* 86.4%

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

       6. *-commutative 86.4%

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

       7. div-inv 86.5%

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

       8. associate-/r* 86.5%

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

   13. Applied egg-rr 86.5%

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

   if -2.40000000000000017e38 < y < 9.8000000000000005e35

   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%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      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. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.1%

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

      3. pow1 54.1%

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

      4. frac-times 54.1%

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

      5. metadata-eval 54.1%

         \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{0.012345679012345678}}{x \cdot x}\right)}^{1}} \]

      6. metadata-eval 54.1%

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

      7. frac-times 54.1%

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

      8. pow-prod-down 54.1%

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

      9. pow-prod-up 54.1%

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

      10. metadata-eval 54.1%

          \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

      11. associate-/r* 54.1%

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

      12. *-commutative 54.1%

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

      13. pow-plus 54.1%

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

      14. pow1 54.1%

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

      15. sqrt-unprod 54.1%

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

      16. add-sqr-sqrt 54.1%

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

      17. frac-2neg 54.1%

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

      18. metadata-eval 54.1%

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

      19. div-inv 54.1%

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

      20. distribute-rgt-neg-in 54.1%

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

      21. metadata-eval 54.1%

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

      22. metadata-eval 54.1%

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

      23. div-inv 54.1%

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

      24. clear-num 54.1%

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

   7. Applied egg-rr 54.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 76.4%

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

      3. pow1 76.4%

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

      4. frac-times 76.4%

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

      5. metadata-eval 76.4%

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

      6. metadata-eval 76.4%

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

      7. frac-times 76.4%

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

      8. pow-prod-down 76.4%

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

      9. pow-prod-up 76.4%

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

      10. metadata-eval 76.4%

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

      11. associate-/r* 76.4%

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

      12. *-commutative 76.4%

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

      13. pow-plus 76.4%

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

      14. pow1 76.4%

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

      15. sqrt-unprod 99.1%

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

      16. add-sqr-sqrt 99.4%

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

      17. inv-pow 99.4%

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

   9. Applied egg-rr 99.4%

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

   if 9.8000000000000005e35 < y

   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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      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. *-commutative 99.5%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 92.5%

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

      1. associate-*r* 92.4%

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

   7. Simplified 92.4%

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

      1. sqrt-div 92.4%

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

      2. metadata-eval 92.4%

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

      3. div-inv 92.6%

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

      4. expm1-log1p-u 38.0%

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

      5. expm1-udef 6.0%

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

   9. Applied egg-rr 6.0%

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

       1. expm1-def 38.0%

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

       2. expm1-log1p 92.6%

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

   11. Simplified 92.6%

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

       1. associate-*l/ 92.5%

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

       2. *-un-lft-identity 92.5%

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

       3. times-frac 92.6%

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

       4. metadata-eval 92.6%

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

       5. metadata-eval 92.6%

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

       6. times-frac 92.5%

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

       7. *-un-lft-identity 92.5%

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

       8. associate-/r* 92.6%

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

   13. Applied egg-rr 92.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+38}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+35}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% 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 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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. *-commutative 99.7%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   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. Add Preprocessing
5. 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%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 9: 99.6% 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 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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. *-commutative 99.7%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   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. Add Preprocessing

5. Final simplification 99.6%

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

Alternative 10: 66.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{-0.1111111111111111}{x}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{-1 + \left(\frac{0.1111111111111111}{x} \cdot -0.1111111111111111\right) \cdot \frac{1}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ -0.1111111111111111 x))))
   (if (<= y -5.8e+134)
     (/
      (+ -1.0 (* (* (/ 0.1111111111111111 x) -0.1111111111111111) (/ 1.0 x)))
      t_0)
     (if (<= y 5.5e+139)
       (+ 1.0 (/ -0.1111111111111111 x))
       (/
        (+ -1.0 (* (/ 0.1111111111111111 x) (/ 0.1111111111111111 x)))
        t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 + (-0.1111111111111111 / x);
	double tmp;
	if (y <= -5.8e+134) {
		tmp = (-1.0 + (((0.1111111111111111 / x) * -0.1111111111111111) * (1.0 / x))) / t_0;
	} else if (y <= 5.5e+139) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (-1.0 + ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + ((-0.1111111111111111d0) / x)
    if (y <= (-5.8d+134)) then
        tmp = ((-1.0d0) + (((0.1111111111111111d0 / x) * (-0.1111111111111111d0)) * (1.0d0 / x))) / t_0
    else if (y <= 5.5d+139) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = ((-1.0d0) + ((0.1111111111111111d0 / x) * (0.1111111111111111d0 / x))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 + (-0.1111111111111111 / x);
	double tmp;
	if (y <= -5.8e+134) {
		tmp = (-1.0 + (((0.1111111111111111 / x) * -0.1111111111111111) * (1.0 / x))) / t_0;
	} else if (y <= 5.5e+139) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (-1.0 + ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + (-0.1111111111111111 / x)
	tmp = 0
	if y <= -5.8e+134:
		tmp = (-1.0 + (((0.1111111111111111 / x) * -0.1111111111111111) * (1.0 / x))) / t_0
	elif y <= 5.5e+139:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (-1.0 + ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(-0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -5.8e+134)
		tmp = Float64(Float64(-1.0 + Float64(Float64(Float64(0.1111111111111111 / x) * -0.1111111111111111) * Float64(1.0 / x))) / t_0);
	elseif (y <= 5.5e+139)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(-1.0 + Float64(Float64(0.1111111111111111 / x) * Float64(0.1111111111111111 / x))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 + (-0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= -5.8e+134)
		tmp = (-1.0 + (((0.1111111111111111 / x) * -0.1111111111111111) * (1.0 / x))) / t_0;
	elseif (y <= 5.5e+139)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (-1.0 + ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.8e+134], N[(N[(-1.0 + N[(N[(N[(0.1111111111111111 / x), $MachinePrecision] * -0.1111111111111111), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 5.5e+139], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(N[(0.1111111111111111 / x), $MachinePrecision] * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.80000000000000023e134

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r- 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r- 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r/ 99.3%

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

      10. fma-neg 99.3%

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

      11. associate-/r* 99.1%

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

      12. metadata-eval 99.1%

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

      13. *-commutative 99.1%

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

      14. associate-/r* 99.1%

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

      15. distribute-neg-frac 99.1%

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

      16. metadata-eval 99.1%

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

      17. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 3.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 15.6%

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

      3. pow1 15.6%

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

      4. frac-times 15.6%

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

      5. metadata-eval 15.6%

         \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{0.012345679012345678}}{x \cdot x}\right)}^{1}} \]

      6. metadata-eval 15.6%

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

      7. frac-times 15.6%

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

      8. pow-prod-down 15.6%

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

      9. pow-prod-up 15.6%

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

      10. metadata-eval 15.6%

          \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

      11. associate-/r* 15.6%

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

      12. *-commutative 15.6%

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

      13. pow-plus 15.6%

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

      14. pow1 15.6%

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

      15. sqrt-unprod 5.5%

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

      16. add-sqr-sqrt 5.5%

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

      17. frac-2neg 5.5%

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

      18. metadata-eval 5.5%

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

      19. div-inv 5.5%

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

      20. distribute-rgt-neg-in 5.5%

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

      21. metadata-eval 5.5%

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

      22. metadata-eval 5.5%

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

      23. div-inv 5.5%

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

      24. clear-num 5.5%

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

   7. Applied egg-rr 5.5%

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

      1. +-commutative 5.5%

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

      2. flip-+ 15.6%

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

      3. swap-sqr 15.6%

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

      4. metadata-eval 15.6%

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

      5. *-un-lft-identity 15.6%

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

      6. frac-times 15.6%

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

      7. metadata-eval 15.6%

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

      8. pow2 15.6%

         \[\leadsto \frac{\frac{0.012345679012345678}{\color{blue}{{x}^{2}}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      9. metadata-eval 15.6%

         \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - \color{blue}{1}}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      10. add-sqr-sqrt 15.6%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{-1 \cdot \frac{-0.1111111111111111}{x}} \cdot \sqrt{-1 \cdot \frac{-0.1111111111111111}{x}}} - 1} \]

      11. sqrt-unprod 4.1%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right) \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right)}} - 1} \]

      12. mul-1-neg 4.1%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)} \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right)} - 1} \]

      13. mul-1-neg 4.1%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\left(-\frac{-0.1111111111111111}{x}\right) \cdot \color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)}} - 1} \]

      14. sqr-neg 4.1%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\color{blue}{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} - 1} \]

      15. sqrt-unprod 0.0%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} - 1} \]

      16. add-sqr-sqrt 3.4%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\frac{-0.1111111111111111}{x}} - 1} \]

   9. Applied egg-rr 3.4%

      \[\leadsto \color{blue}{\frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\frac{-0.1111111111111111}{x} - 1}} \]
   10. Step-by-step derivation

       1. metadata-eval 3.4%

          \[\leadsto \frac{\frac{\color{blue}{-0.1111111111111111 \cdot -0.1111111111111111}}{{x}^{2}} - 1}{\frac{-0.1111111111111111}{x} - 1} \]

       2. unpow2 3.4%

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

       3. frac-times 3.4%

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

       4. div-inv 3.4%

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

       5. associate-*r* 3.4%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 15.6%

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

       8. frac-times 15.6%

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

       9. metadata-eval 15.6%

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

       10. metadata-eval 15.6%

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

       11. frac-times 15.6%

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

       12. sqrt-unprod 15.6%

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

       13. add-sqr-sqrt 15.6%

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

   11. Applied egg-rr 15.6%

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

   if -5.80000000000000023e134 < y < 5.4999999999999996e139

   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%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      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. *-commutative 99.8%

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

      9. associate-*r/ 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 85.2%

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

   if 5.4999999999999996e139 < 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. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      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. *-commutative 99.6%

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

      9. associate-*r/ 99.6%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 4.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.7%

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

      3. pow1 0.7%

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

      4. frac-times 0.7%

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

      5. metadata-eval 0.7%

         \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{0.012345679012345678}}{x \cdot x}\right)}^{1}} \]

      6. metadata-eval 0.7%

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

      7. frac-times 0.7%

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

      8. pow-prod-down 0.7%

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

      9. pow-prod-up 0.7%

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

      10. metadata-eval 0.7%

          \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

      11. associate-/r* 0.7%

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

      12. *-commutative 0.7%

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

      13. pow-plus 0.7%

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

      14. pow1 0.7%

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

      15. sqrt-unprod 0.7%

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

      16. add-sqr-sqrt 0.7%

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

      17. frac-2neg 0.7%

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

      18. metadata-eval 0.7%

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

      19. div-inv 0.7%

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

      20. distribute-rgt-neg-in 0.7%

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

      21. metadata-eval 0.7%

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

      22. metadata-eval 0.7%

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

      23. div-inv 0.7%

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

      24. clear-num 0.7%

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

   7. Applied egg-rr 0.7%

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

      1. +-commutative 0.7%

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

      2. flip-+ 0.7%

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

      3. swap-sqr 0.7%

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

      4. metadata-eval 0.7%

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

      5. *-un-lft-identity 0.7%

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

      6. frac-times 0.7%

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

      7. metadata-eval 0.7%

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

      8. pow2 0.7%

         \[\leadsto \frac{\frac{0.012345679012345678}{\color{blue}{{x}^{2}}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      9. metadata-eval 0.7%

         \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - \color{blue}{1}}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      10. add-sqr-sqrt 0.7%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{-1 \cdot \frac{-0.1111111111111111}{x}} \cdot \sqrt{-1 \cdot \frac{-0.1111111111111111}{x}}} - 1} \]

      11. sqrt-unprod 0.7%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right) \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right)}} - 1} \]

      12. mul-1-neg 0.7%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)} \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right)} - 1} \]

      13. mul-1-neg 0.7%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\left(-\frac{-0.1111111111111111}{x}\right) \cdot \color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)}} - 1} \]

      14. sqr-neg 0.7%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\color{blue}{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} - 1} \]

      15. sqrt-unprod 0.0%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} - 1} \]

      16. add-sqr-sqrt 25.8%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\frac{-0.1111111111111111}{x}} - 1} \]

   9. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\frac{-0.1111111111111111}{x} - 1}} \]
   10. Step-by-step derivation

       1. metadata-eval 25.8%

          \[\leadsto \frac{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{{x}^{2}} - 1}{\frac{-0.1111111111111111}{x} - 1} \]

       2. unpow2 25.8%

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

       3. frac-times 25.8%

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

   11. Applied egg-rr 25.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{-1 + \left(\frac{0.1111111111111111}{x} \cdot -0.1111111111111111\right) \cdot \frac{1}{x}}{-1 + \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{-1 + \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{-1 + \frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e+139)
   (+ 1.0 (/ -0.1111111111111111 x))
   (/
    (+ -1.0 (* (/ 0.1111111111111111 x) (/ 0.1111111111111111 x)))
    (+ -1.0 (/ -0.1111111111111111 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.5e+139) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (-1.0 + ((0.1111111111111111 / x) * (0.1111111111111111 / 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 <= 5.5d+139) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = ((-1.0d0) + ((0.1111111111111111d0 / x) * (0.1111111111111111d0 / 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 <= 5.5e+139) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (-1.0 + ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (-1.0 + (-0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.5e+139:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (-1.0 + ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (-1.0 + (-0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.5e+139)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(-1.0 + Float64(Float64(0.1111111111111111 / x) * Float64(0.1111111111111111 / x))) / Float64(-1.0 + Float64(-0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e+139)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (-1.0 + ((0.1111111111111111 / x) * (0.1111111111111111 / x))) / (-1.0 + (-0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.5e+139], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(N[(0.1111111111111111 / x), $MachinePrecision] * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4999999999999996e139

   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%

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

      4. associate-+r- 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r- 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-*r/ 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 71.2%

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

   if 5.4999999999999996e139 < 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. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      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. *-commutative 99.6%

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

      9. associate-*r/ 99.6%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. distribute-neg-frac 99.6%

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

      16. metadata-eval 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 4.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.7%

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

      3. pow1 0.7%

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

      4. frac-times 0.7%

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

      5. metadata-eval 0.7%

         \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{0.012345679012345678}}{x \cdot x}\right)}^{1}} \]

      6. metadata-eval 0.7%

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

      7. frac-times 0.7%

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

      8. pow-prod-down 0.7%

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

      9. pow-prod-up 0.7%

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

      10. metadata-eval 0.7%

          \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

      11. associate-/r* 0.7%

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

      12. *-commutative 0.7%

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

      13. pow-plus 0.7%

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

      14. pow1 0.7%

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

      15. sqrt-unprod 0.7%

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

      16. add-sqr-sqrt 0.7%

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

      17. frac-2neg 0.7%

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

      18. metadata-eval 0.7%

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

      19. div-inv 0.7%

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

      20. distribute-rgt-neg-in 0.7%

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

      21. metadata-eval 0.7%

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

      22. metadata-eval 0.7%

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

      23. div-inv 0.7%

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

      24. clear-num 0.7%

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

   7. Applied egg-rr 0.7%

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

      1. +-commutative 0.7%

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

      2. flip-+ 0.7%

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

      3. swap-sqr 0.7%

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

      4. metadata-eval 0.7%

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

      5. *-un-lft-identity 0.7%

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

      6. frac-times 0.7%

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

      7. metadata-eval 0.7%

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

      8. pow2 0.7%

         \[\leadsto \frac{\frac{0.012345679012345678}{\color{blue}{{x}^{2}}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      9. metadata-eval 0.7%

         \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - \color{blue}{1}}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      10. add-sqr-sqrt 0.7%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{-1 \cdot \frac{-0.1111111111111111}{x}} \cdot \sqrt{-1 \cdot \frac{-0.1111111111111111}{x}}} - 1} \]

      11. sqrt-unprod 0.7%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right) \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right)}} - 1} \]

      12. mul-1-neg 0.7%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)} \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right)} - 1} \]

      13. mul-1-neg 0.7%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\left(-\frac{-0.1111111111111111}{x}\right) \cdot \color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)}} - 1} \]

      14. sqr-neg 0.7%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\color{blue}{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} - 1} \]

      15. sqrt-unprod 0.0%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} - 1} \]

      16. add-sqr-sqrt 25.8%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\frac{-0.1111111111111111}{x}} - 1} \]

   9. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\frac{-0.1111111111111111}{x} - 1}} \]
   10. Step-by-step derivation

       1. metadata-eval 25.8%

          \[\leadsto \frac{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{{x}^{2}} - 1}{\frac{-0.1111111111111111}{x} - 1} \]

       2. unpow2 25.8%

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

       3. frac-times 25.8%

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

   11. Applied egg-rr 25.8%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+139}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{-1 + \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.3% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		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 <= 0.11d0) 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 <= 0.11) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		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 <= 0.11)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      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. *-commutative 99.6%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 60.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 4.6%

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

      3. pow1 4.6%

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

      4. frac-times 4.6%

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

      5. metadata-eval 4.6%

         \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{0.012345679012345678}}{x \cdot x}\right)}^{1}} \]

      6. metadata-eval 4.6%

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

      7. frac-times 4.6%

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

      8. pow-prod-down 4.6%

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

      9. pow-prod-up 4.6%

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

      10. metadata-eval 4.6%

          \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

      11. associate-/r* 4.6%

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

      12. *-commutative 4.6%

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

      13. pow-plus 4.6%

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

      14. pow1 4.6%

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

      15. sqrt-unprod 1.6%

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

      16. add-sqr-sqrt 1.6%

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

      17. frac-2neg 1.6%

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

      18. metadata-eval 1.6%

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

      19. div-inv 1.6%

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

      20. distribute-rgt-neg-in 1.6%

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

      21. metadata-eval 1.6%

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

      22. metadata-eval 1.6%

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

      23. div-inv 1.6%

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

      24. clear-num 1.6%

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

   7. Applied egg-rr 1.6%

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

      1. +-commutative 1.6%

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

      2. flip-+ 4.6%

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

      3. swap-sqr 4.6%

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

      4. metadata-eval 4.6%

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

      5. *-un-lft-identity 4.6%

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

      6. frac-times 4.6%

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

      7. metadata-eval 4.6%

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

      8. pow2 4.6%

         \[\leadsto \frac{\frac{0.012345679012345678}{\color{blue}{{x}^{2}}} - 1 \cdot 1}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      9. metadata-eval 4.6%

         \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - \color{blue}{1}}{-1 \cdot \frac{-0.1111111111111111}{x} - 1} \]

      10. add-sqr-sqrt 4.6%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{-1 \cdot \frac{-0.1111111111111111}{x}} \cdot \sqrt{-1 \cdot \frac{-0.1111111111111111}{x}}} - 1} \]

      11. sqrt-unprod 1.1%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{\left(-1 \cdot \frac{-0.1111111111111111}{x}\right) \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right)}} - 1} \]

      12. mul-1-neg 1.1%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)} \cdot \left(-1 \cdot \frac{-0.1111111111111111}{x}\right)} - 1} \]

      13. mul-1-neg 1.1%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\left(-\frac{-0.1111111111111111}{x}\right) \cdot \color{blue}{\left(-\frac{-0.1111111111111111}{x}\right)}} - 1} \]

      14. sqr-neg 1.1%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\sqrt{\color{blue}{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} - 1} \]

      15. sqrt-unprod 0.0%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} - 1} \]

      16. add-sqr-sqrt 33.5%

          \[\leadsto \frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\color{blue}{\frac{-0.1111111111111111}{x}} - 1} \]

   9. Applied egg-rr 33.5%

      \[\leadsto \color{blue}{\frac{\frac{0.012345679012345678}{{x}^{2}} - 1}{\frac{-0.1111111111111111}{x} - 1}} \]

   10. Taylor expanded in x around 0 60.4%

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

   if 0.110000000000000001 < x

   1. Initial program 99.7%

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

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

      4. associate-+r- 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r- 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 63.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 63.2%

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

      3. pow1 63.2%

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

      4. frac-times 63.2%

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

      5. metadata-eval 63.2%

         \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{0.012345679012345678}}{x \cdot x}\right)}^{1}} \]

      6. metadata-eval 63.2%

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

      7. frac-times 63.2%

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

      8. pow-prod-down 63.2%

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

      9. pow-prod-up 63.2%

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

      10. metadata-eval 63.2%

          \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

      11. associate-/r* 63.2%

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

      12. *-commutative 63.2%

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

      13. pow-plus 63.2%

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

      14. pow1 63.2%

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

      15. sqrt-unprod 63.2%

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

      16. add-sqr-sqrt 63.2%

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

      17. frac-2neg 63.2%

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

      18. metadata-eval 63.2%

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

      19. div-inv 63.2%

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

      20. distribute-rgt-neg-in 63.2%

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

      21. metadata-eval 63.2%

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

      22. metadata-eval 63.2%

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

      23. div-inv 63.2%

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

      24. clear-num 63.2%

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

   7. Applied egg-rr 63.2%

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

   8. Taylor expanded in x around inf 63.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.4% 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 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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%

      \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

   4. associate-+r- 99.7%

      \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

   5. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. *-commutative 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   9. associate-*r/ 99.7%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fma-neg 99.6%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   11. associate-/r* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   12. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   17. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 62.3%

   \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]

6. Final simplification 62.3%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]
7. Add Preprocessing

Alternative 14: 31.6% 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 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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%

      \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

   4. associate-+r- 99.7%

      \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

   5. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. *-commutative 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   9. associate-*r/ 99.7%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fma-neg 99.6%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   11. associate-/r* 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   12. metadata-eval 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   13. *-commutative 99.7%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   14. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   15. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   17. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 62.3%

   \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

      \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} \]

   2. sqrt-unprod 34.6%

      \[\leadsto 1 + \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} \]

   3. pow1 34.6%

      \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}\right)}^{1}}} \]

   4. frac-times 34.6%

      \[\leadsto 1 + \sqrt{{\color{blue}{\left(\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}\right)}}^{1}} \]

   5. metadata-eval 34.6%

      \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{0.012345679012345678}}{x \cdot x}\right)}^{1}} \]

   6. metadata-eval 34.6%

      \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}\right)}^{1}} \]

   7. frac-times 34.6%

      \[\leadsto 1 + \sqrt{{\color{blue}{\left(\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}\right)}}^{1}} \]

   8. pow-prod-down 34.6%

      \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{1} \cdot {\left(\frac{0.1111111111111111}{x}\right)}^{1}}} \]

   9. pow-prod-up 34.6%

      \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{\left(1 + 1\right)}}} \]

   10. metadata-eval 34.6%

       \[\leadsto 1 + \sqrt{{\left(\frac{\color{blue}{\frac{1}{9}}}{x}\right)}^{\left(1 + 1\right)}} \]

   11. associate-/r* 34.6%

       \[\leadsto 1 + \sqrt{{\color{blue}{\left(\frac{1}{9 \cdot x}\right)}}^{\left(1 + 1\right)}} \]

   12. *-commutative 34.6%

       \[\leadsto 1 + \sqrt{{\left(\frac{1}{\color{blue}{x \cdot 9}}\right)}^{\left(1 + 1\right)}} \]

   13. pow-plus 34.6%

       \[\leadsto 1 + \sqrt{\color{blue}{{\left(\frac{1}{x \cdot 9}\right)}^{1} \cdot \frac{1}{x \cdot 9}}} \]

   14. pow1 34.6%

       \[\leadsto 1 + \sqrt{\color{blue}{\frac{1}{x \cdot 9}} \cdot \frac{1}{x \cdot 9}} \]

   15. sqrt-unprod 33.1%

       \[\leadsto 1 + \color{blue}{\sqrt{\frac{1}{x \cdot 9}} \cdot \sqrt{\frac{1}{x \cdot 9}}} \]

   16. add-sqr-sqrt 33.1%

       \[\leadsto 1 + \color{blue}{\frac{1}{x \cdot 9}} \]

   17. frac-2neg 33.1%

       \[\leadsto 1 + \color{blue}{\frac{-1}{-x \cdot 9}} \]

   18. metadata-eval 33.1%

       \[\leadsto 1 + \frac{\color{blue}{-1}}{-x \cdot 9} \]

   19. div-inv 33.1%

       \[\leadsto 1 + \color{blue}{-1 \cdot \frac{1}{-x \cdot 9}} \]

   20. distribute-rgt-neg-in 33.1%

       \[\leadsto 1 + -1 \cdot \frac{1}{\color{blue}{x \cdot \left(-9\right)}} \]

   21. metadata-eval 33.1%

       \[\leadsto 1 + -1 \cdot \frac{1}{x \cdot \color{blue}{-9}} \]

   22. metadata-eval 33.1%

       \[\leadsto 1 + -1 \cdot \frac{1}{x \cdot \color{blue}{\frac{1}{-0.1111111111111111}}} \]

   23. div-inv 33.1%

       \[\leadsto 1 + -1 \cdot \frac{1}{\color{blue}{\frac{x}{-0.1111111111111111}}} \]

   24. clear-num 33.1%

       \[\leadsto 1 + -1 \cdot \color{blue}{\frac{-0.1111111111111111}{x}} \]

7. Applied egg-rr 33.1%

   \[\leadsto 1 + \color{blue}{-1 \cdot \frac{-0.1111111111111111}{x}} \]

8. Taylor expanded in x around inf 33.1%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 33.1%

   \[\leadsto 1 \]
10. Add Preprocessing

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 2024036 
(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)))))