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

Percentage Accurate: 99.7% → 99.7%

Time: 10.3s

Alternatives: 16

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

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

FPCore

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

C

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

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 / sqrt((x * 9.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. metadata-eval 99.6%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

4. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+42} \lor \neg \left(y \leq 1.02 \cdot 10^{+36}\right):\\ \;\;\;\;1 - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.9e+42) (not (<= y 1.02e+36)))
   (- 1.0 (* 0.3333333333333333 (/ y (sqrt x))))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e+42) || !(y <= 1.02e+36)) {
		tmp = 1.0 - (0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 - pow((x * 9.0), -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 ((y <= (-1.9d+42)) .or. (.not. (y <= 1.02d+36))) then
        tmp = 1.0d0 - (0.3333333333333333d0 * (y / sqrt(x)))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e+42) || !(y <= 1.02e+36)) {
		tmp = 1.0 - (0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.9e+42) or not (y <= 1.02e+36):
		tmp = 1.0 - (0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.9e+42) || !(y <= 1.02e+36))
		tmp = Float64(1.0 - Float64(0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.9e+42) || ~((y <= 1.02e+36)))
		tmp = 1.0 - (0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.9e+42], N[Not[LessEqual[y, 1.02e+36]], $MachinePrecision]], N[(1.0 - N[(0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8999999999999999e42 or 1.02000000000000003e36 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 94.2%

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

      1. *-commutative 94.2%

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

      2. sqrt-div 94.1%

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

      3. metadata-eval 94.1%

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

      4. div-inv 94.2%

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

   5. Applied egg-rr 94.2%

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

   if -1.8999999999999999e42 < y < 1.02000000000000003e36

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      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-/l* 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 98.0%

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

      1. metadata-eval 98.0%

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

      2. distribute-neg-frac 98.0%

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

      3. add-sqr-sqrt 97.8%

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

      4. sqrt-unprod 71.2%

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

      5. frac-times 71.2%

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

      6. metadata-eval 71.2%

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

      7. metadata-eval 71.2%

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

      8. frac-times 71.2%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 39.4%

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

   7. Applied egg-rr 39.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.2%

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

      3. frac-times 71.2%

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

      4. metadata-eval 71.2%

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

      5. metadata-eval 71.2%

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

      6. frac-times 71.2%

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

      7. sqrt-unprod 97.8%

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

      8. add-sqr-sqrt 98.0%

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

      9. clear-num 98.0%

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

      10. div-inv 98.1%

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

      11. metadata-eval 98.1%

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

      12. inv-pow 98.1%

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

   9. Applied egg-rr 98.1%

      \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+42} \lor \neg \left(y \leq 1.02 \cdot 10^{+36}\right):\\ \;\;\;\;1 - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+38} \lor \neg \left(y \leq 2.75 \cdot 10^{+39}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.8e+38) (not (<= y 2.75e+39)))
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.8e+38) || !(y <= 2.75e+39)) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else {
		tmp = 1.0 - pow((x * 9.0), -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 ((y <= (-3.8d+38)) .or. (.not. (y <= 2.75d+39))) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.8e+38) || !(y <= 2.75e+39)) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.8e+38) or not (y <= 2.75e+39):
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.8e+38) || !(y <= 2.75e+39))
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.8e+38) || ~((y <= 2.75e+39)))
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.8e+38], N[Not[LessEqual[y, 2.75e+39]], $MachinePrecision]], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999998e38 or 2.7499999999999999e39 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 94.2%

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

      1. *-commutative 94.2%

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

      2. sqrt-div 94.1%

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

      3. metadata-eval 94.1%

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

      4. div-inv 94.2%

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

      5. clear-num 94.1%

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

   5. Applied egg-rr 94.1%

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

      1. un-div-inv 94.2%

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

      2. associate-/r/ 94.1%

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

      3. metadata-eval 94.1%

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

      4. sqrt-div 94.3%

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

      5. clear-num 94.2%

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

      6. sqrt-div 94.2%

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

      7. metadata-eval 94.2%

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

      8. div-inv 94.2%

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

      9. metadata-eval 94.2%

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

      10. associate-/r/ 94.2%

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

      11. clear-num 94.3%

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

   7. Applied egg-rr 94.3%

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

   if -3.7999999999999998e38 < y < 2.7499999999999999e39

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      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-/l* 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 98.0%

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

      1. metadata-eval 98.0%

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

      2. distribute-neg-frac 98.0%

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

      3. add-sqr-sqrt 97.8%

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

      4. sqrt-unprod 71.2%

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

      5. frac-times 71.2%

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

      6. metadata-eval 71.2%

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

      7. metadata-eval 71.2%

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

      8. frac-times 71.2%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 39.4%

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

   7. Applied egg-rr 39.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.2%

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

      3. frac-times 71.2%

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

      4. metadata-eval 71.2%

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

      5. metadata-eval 71.2%

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

      6. frac-times 71.2%

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

      7. sqrt-unprod 97.8%

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

      8. add-sqr-sqrt 98.0%

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

      9. clear-num 98.0%

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

      10. div-inv 98.1%

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

      11. metadata-eval 98.1%

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

      12. inv-pow 98.1%

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

   9. Applied egg-rr 98.1%

      \[\leadsto 1 + \left(-\color{blue}{{\left(x \cdot 9\right)}^{-1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+38} \lor \neg \left(y \leq 2.75 \cdot 10^{+39}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+42}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e+42)
   (- 1.0 (/ 0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 4.2e+36)
     (- 1.0 (pow (* x 9.0) -1.0))
     (- 1.0 (* 0.3333333333333333 (/ y (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.3e+42) {
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 4.2e+36) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - (0.3333333333333333 * (y / sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.3d+42)) then
        tmp = 1.0d0 - (0.3333333333333333d0 / (sqrt(x) / y))
    else if (y <= 4.2d+36) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 - (0.3333333333333333d0 * (y / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.3e+42) {
		tmp = 1.0 - (0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 4.2e+36) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - (0.3333333333333333 * (y / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.3e+42:
		tmp = 1.0 - (0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 4.2e+36:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 - (0.3333333333333333 * (y / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.3e+42)
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 4.2e+36)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 - Float64(0.3333333333333333 * Float64(y / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.3e+42)
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 4.2e+36)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 - (0.3333333333333333 * (y / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.3e+42], N[(1.0 - N[(0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+36], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2999999999999999e42

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. *-commutative 93.6%

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

      2. sqrt-div 93.5%

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

      3. metadata-eval 93.5%

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

      4. div-inv 93.6%

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

      5. clear-num 93.4%

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

   5. Applied egg-rr 93.4%

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

      1. un-div-inv 93.7%

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

   7. Applied egg-rr 93.7%

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

   if -3.2999999999999999e42 < y < 4.20000000000000009e36

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      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-/l* 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 98.0%

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

      1. metadata-eval 98.0%

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

      2. distribute-neg-frac 98.0%

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

      3. add-sqr-sqrt 97.8%

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

      4. sqrt-unprod 71.2%

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

      5. frac-times 71.2%

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

      6. metadata-eval 71.2%

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

      7. metadata-eval 71.2%

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

      8. frac-times 71.2%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 39.4%

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

   7. Applied egg-rr 39.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.2%

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

      3. frac-times 71.2%

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

      4. metadata-eval 71.2%

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

      5. metadata-eval 71.2%

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

      6. frac-times 71.2%

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

      7. sqrt-unprod 97.8%

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

      8. add-sqr-sqrt 98.0%

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

      9. clear-num 98.0%

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

      10. div-inv 98.1%

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

      11. metadata-eval 98.1%

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

      12. inv-pow 98.1%

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

   9. Applied egg-rr 98.1%

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

   if 4.20000000000000009e36 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 94.7%

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

      1. *-commutative 94.7%

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

      2. sqrt-div 94.7%

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

      3. metadata-eval 94.7%

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

      4. div-inv 94.8%

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

   5. Applied egg-rr 94.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+42}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+41}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+37}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.26e+41)
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (if (<= y 2.6e+37)
     (- 1.0 (pow (* x 9.0) -1.0))
     (- 1.0 (/ (/ y (sqrt x)) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.26e+41) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else if (y <= 2.6e+37) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.26d+41)) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else if (y <= 2.6d+37) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 - ((y / sqrt(x)) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.26e+41) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else if (y <= 2.6e+37) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - ((y / Math.sqrt(x)) / 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.26e+41:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	elif y <= 2.6e+37:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 - ((y / math.sqrt(x)) / 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.26e+41)
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	elseif (y <= 2.6e+37)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 - Float64(Float64(y / sqrt(x)) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.26e+41)
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	elseif (y <= 2.6e+37)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.26e+41], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+37], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.26000000000000001e41

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 93.6%

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

      1. *-commutative 93.6%

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

      2. sqrt-div 93.5%

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

      3. metadata-eval 93.5%

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

      4. div-inv 93.6%

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

      5. clear-num 93.4%

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

   5. Applied egg-rr 93.4%

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

      1. un-div-inv 93.7%

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

      2. associate-/r/ 93.5%

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

      3. metadata-eval 93.5%

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

      4. sqrt-div 93.7%

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

      5. clear-num 93.6%

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

      6. sqrt-div 93.7%

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

      7. metadata-eval 93.7%

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

      8. div-inv 93.8%

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

      9. metadata-eval 93.8%

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

      10. associate-/r/ 93.8%

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

      11. clear-num 93.8%

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

   7. Applied egg-rr 93.8%

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

   if -1.26000000000000001e41 < y < 2.5999999999999999e37

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      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-/l* 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 98.0%

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

      1. metadata-eval 98.0%

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

      2. distribute-neg-frac 98.0%

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

      3. add-sqr-sqrt 97.8%

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

      4. sqrt-unprod 71.2%

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

      5. frac-times 71.2%

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

      6. metadata-eval 71.2%

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

      7. metadata-eval 71.2%

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

      8. frac-times 71.2%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 39.4%

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

   7. Applied egg-rr 39.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 71.2%

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

      3. frac-times 71.2%

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

      4. metadata-eval 71.2%

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

      5. metadata-eval 71.2%

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

      6. frac-times 71.2%

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

      7. sqrt-unprod 97.8%

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

      8. add-sqr-sqrt 98.0%

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

      9. clear-num 98.0%

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

      10. div-inv 98.1%

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

      11. metadata-eval 98.1%

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

      12. inv-pow 98.1%

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

   9. Applied egg-rr 98.1%

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

   if 2.5999999999999999e37 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 94.7%

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

      1. *-commutative 94.7%

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

      2. sqrt-div 94.7%

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

      3. metadata-eval 94.7%

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

      4. div-inv 94.8%

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

      5. clear-num 94.7%

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

   5. Applied egg-rr 94.7%

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

      1. un-div-inv 94.7%

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

      2. associate-/r/ 94.6%

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

      3. metadata-eval 94.6%

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

      4. sqrt-div 94.9%

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

      5. clear-num 94.8%

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

      6. sqrt-div 94.6%

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

      7. metadata-eval 94.6%

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

      8. div-inv 94.7%

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

      9. metadata-eval 94.7%

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

      10. associate-/r/ 94.7%

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

      11. clear-num 94.9%

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

      12. sqrt-prod 94.7%

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

      13. associate-/r* 94.9%

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

      14. metadata-eval 94.9%

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

   7. Applied egg-rr 94.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+41}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+37}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+81}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e+97)
   (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))
   (if (<= y 4.1e+81)
     (- 1.0 (pow (* x 9.0) -1.0))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+97) {
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} else if (y <= 4.1e+81) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = (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 <= (-4.2d+97)) then
        tmp = sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0))
    else if (y <= 4.1d+81) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+97) {
		tmp = Math.sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} else if (y <= 4.1e+81) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e+97:
		tmp = math.sqrt((1.0 / x)) * (y * -0.3333333333333333)
	elif y <= 4.1e+81:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e+97)
		tmp = Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333));
	elseif (y <= 4.1e+81)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.2e+97)
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	elseif (y <= 4.1e+81)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e+97], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+81], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.20000000000000023e97

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 94.7%

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

      1. associate-*r* 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*l* 94.8%

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

   9. Simplified 94.8%

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

   if -4.20000000000000023e97 < y < 4.10000000000000012e81

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      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-/l* 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 94.3%

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

      1. metadata-eval 94.3%

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

      2. distribute-neg-frac 94.3%

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

      3. add-sqr-sqrt 94.1%

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

      4. sqrt-unprod 68.7%

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

      5. frac-times 68.7%

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

      6. metadata-eval 68.7%

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

      7. metadata-eval 68.7%

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

      8. frac-times 68.7%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 40.1%

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

   7. Applied egg-rr 40.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 68.7%

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

      3. frac-times 68.7%

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

      4. metadata-eval 68.7%

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

      5. metadata-eval 68.7%

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

      6. frac-times 68.7%

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

      7. sqrt-unprod 94.1%

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

      8. add-sqr-sqrt 94.3%

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

      9. clear-num 94.3%

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

      10. div-inv 94.3%

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

      11. metadata-eval 94.3%

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

      12. inv-pow 94.3%

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

   9. Applied egg-rr 94.3%

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

   if 4.10000000000000012e81 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around inf 95.4%

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

      1. associate-*r* 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*l* 95.4%

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

   9. Simplified 95.4%

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

       1. *-commutative 95.4%

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

       2. sqrt-div 95.3%

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

       3. metadata-eval 95.3%

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

       4. un-div-inv 95.6%

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

       5. *-commutative 95.6%

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

   11. Applied egg-rr 95.6%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+81}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+97} \lor \neg \left(y \leq 1.15 \cdot 10^{+81}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.3e+97) (not (<= y 1.15e+81)))
   (* y (/ -0.3333333333333333 (sqrt x)))
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.3e+97) || !(y <= 1.15e+81)) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / 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 <= (-3.3d+97)) .or. (.not. (y <= 1.15d+81))) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.3e+97) || !(y <= 1.15e+81)) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.3e+97) or not (y <= 1.15e+81):
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	else:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.3e+97) || !(y <= 1.15e+81))
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.3e+97) || ~((y <= 1.15e+81)))
		tmp = y * (-0.3333333333333333 / sqrt(x));
	else
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.3e+97], N[Not[LessEqual[y, 1.15e+81]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3000000000000001e97 or 1.1499999999999999e81 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 95.1%

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

      1. associate-*r* 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*l* 95.1%

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

   9. Simplified 95.1%

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

       1. *-commutative 95.1%

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

       2. sqrt-div 95.0%

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

       3. metadata-eval 95.0%

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

       4. un-div-inv 95.2%

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

       5. *-commutative 95.2%

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

   11. Applied egg-rr 95.2%

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

       1. associate-/l* 95.0%

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

   13. Simplified 95.0%

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

   if -3.3000000000000001e97 < y < 1.1499999999999999e81

   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. Taylor expanded in y around 0 94.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+97} \lor \neg \left(y \leq 1.15 \cdot 10^{+81}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+97} \lor \neg \left(y \leq 4.8 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.6e+97) (not (<= y 4.8e+82)))
   (/ (* y -0.3333333333333333) (sqrt x))
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.6e+97) || !(y <= 4.8e+82)) {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / 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 <= (-6.6d+97)) .or. (.not. (y <= 4.8d+82))) then
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    else
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.6e+97) || !(y <= 4.8e+82)) {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.6e+97) or not (y <= 4.8e+82):
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	else:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.6e+97) || !(y <= 4.8e+82))
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	else
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.6e+97) || ~((y <= 4.8e+82)))
		tmp = (y * -0.3333333333333333) / sqrt(x);
	else
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.6e+97], N[Not[LessEqual[y, 4.8e+82]], $MachinePrecision]], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.6000000000000003e97 or 4.79999999999999996e82 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 95.1%

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

      1. associate-*r* 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*l* 95.1%

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

   9. Simplified 95.1%

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

       1. *-commutative 95.1%

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

       2. sqrt-div 95.0%

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

       3. metadata-eval 95.0%

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

       4. un-div-inv 95.2%

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

       5. *-commutative 95.2%

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

   11. Applied egg-rr 95.2%

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

   if -6.6000000000000003e97 < y < 4.79999999999999996e82

   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. Taylor expanded in y around 0 94.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+97} \lor \neg \left(y \leq 4.8 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+84}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e+97)
   (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))
   (if (<= y 3.5e+84)
     (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.3e+97) {
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} else if (y <= 3.5e+84) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = (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 <= (-3.3d+97)) then
        tmp = sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0))
    else if (y <= 3.5d+84) then
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.3e+97) {
		tmp = Math.sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} else if (y <= 3.5e+84) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.3e+97:
		tmp = math.sqrt((1.0 / x)) * (y * -0.3333333333333333)
	elif y <= 3.5e+84:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.3e+97)
		tmp = Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333));
	elseif (y <= 3.5e+84)
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.3e+97)
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	elseif (y <= 3.5e+84)
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.3e+97], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+84], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.3000000000000001e97

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 94.7%

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

      1. associate-*r* 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*l* 94.8%

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

   9. Simplified 94.8%

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

   if -3.3000000000000001e97 < y < 3.4999999999999999e84

   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. Taylor expanded in y around 0 94.3%

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

   if 3.4999999999999999e84 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around inf 95.4%

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

      1. associate-*r* 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*l* 95.4%

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

   9. Simplified 95.4%

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

       1. *-commutative 95.4%

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

       2. sqrt-div 95.3%

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

       3. metadata-eval 95.3%

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

       4. un-div-inv 95.6%

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

       5. *-commutative 95.6%

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

   11. Applied egg-rr 95.6%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+84}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (0.1111111111111111 + (y * (0.3333333333333333 * sqrt(x)))) / -x;
	} else {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	}
	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 + (y * (0.3333333333333333d0 * sqrt(x)))) / -x
    else
        tmp = 1.0d0 - ((y / sqrt(x)) / 3.0d0)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = (0.1111111111111111 + (y * (0.3333333333333333 * math.sqrt(x)))) / -x
	else:
		tmp = 1.0 - ((y / math.sqrt(x)) / 3.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = (0.1111111111111111 + (y * (0.3333333333333333 * sqrt(x)))) / -x;
	else
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.5%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   8. Taylor expanded in x around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. associate-*r* 98.3%

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

   10. Simplified 98.3%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 97.8%

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

      1. *-commutative 97.8%

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

      2. sqrt-div 97.8%

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

      3. metadata-eval 97.8%

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

      4. div-inv 97.8%

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

      5. clear-num 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. un-div-inv 97.8%

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

      2. associate-/r/ 97.7%

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

      3. metadata-eval 97.7%

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

      4. sqrt-div 97.8%

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

      5. clear-num 97.8%

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

      6. sqrt-div 97.8%

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

      7. metadata-eval 97.8%

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

      8. div-inv 97.8%

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

      9. metadata-eval 97.8%

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

      10. associate-/r/ 97.8%

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

      11. clear-num 97.9%

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

      12. sqrt-prod 97.9%

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

      13. associate-/r* 97.9%

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

      14. metadata-eval 97.9%

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

   7. Applied egg-rr 97.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{0.1111111111111111 + y \cdot \left(0.3333333333333333 \cdot \sqrt{x}\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	}
	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 = (((-0.3333333333333333d0) * (y * sqrt(x))) - 0.1111111111111111d0) / x
    else
        tmp = 1.0d0 - ((y / sqrt(x)) / 3.0d0)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = ((-0.3333333333333333 * (y * math.sqrt(x))) - 0.1111111111111111) / x
	else:
		tmp = 1.0 - ((y / math.sqrt(x)) / 3.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	else
		tmp = 1.0 - ((y / sqrt(x)) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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. *-commutative 99.5%

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

      3. associate-/r* 99.4%

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

      4. metadata-eval 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. neg-mul-1 99.4%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 98.3%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 97.8%

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

      1. *-commutative 97.8%

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

      2. sqrt-div 97.8%

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

      3. metadata-eval 97.8%

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

      4. div-inv 97.8%

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

      5. clear-num 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. un-div-inv 97.8%

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

      2. associate-/r/ 97.7%

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

      3. metadata-eval 97.7%

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

      4. sqrt-div 97.8%

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

      5. clear-num 97.8%

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

      6. sqrt-div 97.8%

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

      7. metadata-eval 97.8%

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

      8. div-inv 97.8%

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

      9. metadata-eval 97.8%

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

      10. associate-/r/ 97.8%

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

      11. clear-num 97.9%

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

      12. sqrt-prod 97.9%

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

      13. associate-/r* 97.9%

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

      14. metadata-eval 97.9%

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

   7. Applied egg-rr 97.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{\sqrt{x}}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. metadata-eval 99.6%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

4. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

   1. sub-neg 99.7%

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

   2. sub-neg 99.7%

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

   3. associate-+l+ 99.7%

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

   4. metadata-eval 99.7%

      \[\leadsto 1 + \left(\left(-\frac{1}{x \cdot \color{blue}{\frac{1}{0.1111111111111111}}}\right) + \left(-\frac{y}{\sqrt{x \cdot 9}}\right)\right) \]

   5. div-inv 99.6%

      \[\leadsto 1 + \left(\left(-\frac{1}{\color{blue}{\frac{x}{0.1111111111111111}}}\right) + \left(-\frac{y}{\sqrt{x \cdot 9}}\right)\right) \]

   6. clear-num 99.6%

      \[\leadsto 1 + \left(\left(-\color{blue}{\frac{0.1111111111111111}{x}}\right) + \left(-\frac{y}{\sqrt{x \cdot 9}}\right)\right) \]

   7. distribute-neg-frac 99.6%

      \[\leadsto 1 + \left(\color{blue}{\frac{-0.1111111111111111}{x}} + \left(-\frac{y}{\sqrt{x \cdot 9}}\right)\right) \]

   8. metadata-eval 99.6%

      \[\leadsto 1 + \left(\frac{\color{blue}{-0.1111111111111111}}{x} + \left(-\frac{y}{\sqrt{x \cdot 9}}\right)\right) \]

   9. distribute-neg-frac2 99.6%

      \[\leadsto 1 + \left(\frac{-0.1111111111111111}{x} + \color{blue}{\frac{y}{-\sqrt{x \cdot 9}}}\right) \]

   10. sqrt-prod 99.6%

       \[\leadsto 1 + \left(\frac{-0.1111111111111111}{x} + \frac{y}{-\color{blue}{\sqrt{x} \cdot \sqrt{9}}}\right) \]

   11. distribute-rgt-neg-in 99.6%

       \[\leadsto 1 + \left(\frac{-0.1111111111111111}{x} + \frac{y}{\color{blue}{\sqrt{x} \cdot \left(-\sqrt{9}\right)}}\right) \]

   12. metadata-eval 99.6%

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

   13. metadata-eval 99.6%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 13: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 1.0 - ((0.1111111111111111 / x) + ((y * 0.3333333333333333) / 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) + ((y * 0.3333333333333333d0) / sqrt(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. metadata-eval 99.6%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

4. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 99.6%

   \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]
8. Step-by-step derivation

   1. associate--l- 99.6%

      \[\leadsto \color{blue}{1 - \left(\frac{0.1111111111111111}{x} + \frac{y}{\sqrt{x \cdot 9}}\right)} \]

   2. sub-neg 99.6%

      \[\leadsto \color{blue}{1 + \left(-\left(\frac{0.1111111111111111}{x} + \frac{y}{\sqrt{x \cdot 9}}\right)\right)} \]

   3. div-inv 99.6%

      \[\leadsto 1 + \left(-\left(\frac{0.1111111111111111}{x} + \color{blue}{y \cdot \frac{1}{\sqrt{x \cdot 9}}}\right)\right) \]

   4. *-commutative 99.6%

      \[\leadsto 1 + \left(-\left(\frac{0.1111111111111111}{x} + \color{blue}{\frac{1}{\sqrt{x \cdot 9}} \cdot y}\right)\right) \]

   5. metadata-eval 99.6%

      \[\leadsto 1 + \left(-\left(\frac{0.1111111111111111}{x} + \frac{\color{blue}{\sqrt{1}}}{\sqrt{x \cdot 9}} \cdot y\right)\right) \]

   6. metadata-eval 99.6%

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

   7. div-inv 99.6%

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

   8. sqrt-div 99.6%

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

   9. clear-num 99.6%

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

   10. sqrt-div 99.5%

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

   11. metadata-eval 99.5%

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

   12. associate-/r/ 99.6%

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

   13. un-div-inv 99.5%

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

   14. *-commutative 99.5%

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

   15. clear-num 99.6%

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

9. Applied egg-rr 99.6%

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

    1. unsub-neg 99.6%

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

    2. associate-*l/ 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 14: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 0.1111111111111111 x)) (/ y (sqrt (* x 9.0)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) - (y / sqrt((x * 9.0d0)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / Math.sqrt((x * 9.0)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) - (y / math.sqrt((x * 9.0)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) - Float64(y / sqrt(Float64(x * 9.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. metadata-eval 99.6%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

4. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 99.6%

   \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

8. Final simplification 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \]
9. Add Preprocessing

Alternative 15: 61.8% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. 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. *-commutative 99.6%

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

   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. Taylor expanded in y around 0 64.2%

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

6. Final simplification 64.2%

   \[\leadsto 1 + 0.1111111111111111 \cdot \frac{-1}{x} \]
7. Add Preprocessing

Alternative 16: 61.8% 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 99.6%

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

   1. associate--l- 99.6%

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

   2. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. sub-neg 99.6%

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

   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-/l* 99.6%

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

   10. fma-neg 99.6%

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

   11. associate-/r* 99.6%

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

   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 64.2%

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

6. Final simplification 64.2%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]
7. 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 2024059 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))