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

Percentage Accurate: 99.7% → 99.7%

Time: 11.0s

Alternatives: 15

Speedup: N/A×

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

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

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 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   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 99.7%

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

Alternative 2: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+42} \lor \neg \left(y \leq 2.6 \cdot 10^{+16}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.1e+42) (not (<= y 2.6e+16)))
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.1e+42) || !(y <= 2.6e+16)) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-5.1d+42)) .or. (.not. (y <= 2.6d+16))) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5.1e+42) || !(y <= 2.6e+16)) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.1e+42) or not (y <= 2.6e+16):
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.1e+42) || !(y <= 2.6e+16))
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.1e+42) || ~((y <= 2.6e+16)))
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.1e+42], N[Not[LessEqual[y, 2.6e+16]], $MachinePrecision]], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.0999999999999999e42 or 2.6e16 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.6%

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

   4. Taylor expanded in x around inf 91.9%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   5. 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 99.8%

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

      4. pow1/2 99.8%

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

   6. Applied egg-rr 92.0%

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

      1. unpow1/2 99.8%

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

   8. Simplified 92.0%

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

   if -5.0999999999999999e42 < y < 2.6e16

   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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+42} \lor \neg \left(y \leq 2.6 \cdot 10^{+16}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+42} \lor \neg \left(y \leq 2.1 \cdot 10^{+16}\right):\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.5e+42) (not (<= y 2.1e+16)))
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e+42) || !(y <= 2.1e+16)) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.5d+42)) .or. (.not. (y <= 2.1d+16))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e+42) || !(y <= 2.1e+16)) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.5e+42) or not (y <= 2.1e+16):
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.5e+42) || !(y <= 2.1e+16))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.5e+42) || ~((y <= 2.1e+16)))
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.5e+42], N[Not[LessEqual[y, 2.1e+16]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.50000000000000052e42 or 2.1e16 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. *-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 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 92.0%

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

   if -6.50000000000000052e42 < y < 2.1e16

   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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+42} \lor \neg \left(y \leq 2.1 \cdot 10^{+16}\right):\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+42} \lor \neg \left(y \leq 2.6 \cdot 10^{+16}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.5e+42) (not (<= y 2.6e+16)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.5e+42) || !(y <= 2.6e+16)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.5d+42)) .or. (.not. (y <= 2.6d+16))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.5e+42) || !(y <= 2.6e+16)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.5e+42) or not (y <= 2.6e+16):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.5e+42) || !(y <= 2.6e+16))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.5e+42) || ~((y <= 2.6e+16)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.5e+42], N[Not[LessEqual[y, 2.6e+16]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000012e42 or 2.6e16 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. *-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 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 91.8%

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

   if -4.50000000000000012e42 < y < 2.6e16

   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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.7%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+42} \lor \neg \left(y \leq 2.6 \cdot 10^{+16}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+79} \lor \neg \left(y \leq 4 \cdot 10^{+80}\right):\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6e+79) (not (<= y 4e+80)))
   (* (pow x -0.5) (* y -0.3333333333333333))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+79) || !(y <= 4e+80)) {
		tmp = pow(x, -0.5) * (y * -0.3333333333333333);
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.6d+79)) .or. (.not. (y <= 4d+80))) then
        tmp = (x ** (-0.5d0)) * (y * (-0.3333333333333333d0))
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+79) || !(y <= 4e+80)) {
		tmp = Math.pow(x, -0.5) * (y * -0.3333333333333333);
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.6e+79) or not (y <= 4e+80):
		tmp = math.pow(x, -0.5) * (y * -0.3333333333333333)
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.6e+79) || !(y <= 4e+80))
		tmp = Float64((x ^ -0.5) * Float64(y * -0.3333333333333333));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.6e+79) || ~((y <= 4e+80)))
		tmp = (x ^ -0.5) * (y * -0.3333333333333333);
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.6e+79], N[Not[LessEqual[y, 4e+80]], $MachinePrecision]], N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.60000000000000001e79 or 4e80 < y

   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 95.8%

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

      1. associate-*r* 95.7%

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

      2. *-commutative 95.7%

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

      3. associate-*l* 95.8%

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

      4. unpow-1 95.8%

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

      5. metadata-eval 95.8%

         \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-0.3333333333333333 \cdot y\right) \]

      6. pow-sqr 95.8%

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

      7. rem-sqrt-square 95.8%

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

      8. rem-square-sqrt 95.6%

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

      9. fabs-sqr 95.6%

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

      10. rem-square-sqrt 95.8%

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

   9. Simplified 95.8%

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

   if -1.60000000000000001e79 < y < 4e80

   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* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 96.1%

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

      1. *-commutative 96.1%

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

      2. inv-pow 96.1%

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

      3. metadata-eval 96.1%

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

      4. sqrt-pow1 70.0%

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

      5. metadata-eval 70.0%

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

      6. metadata-eval 70.0%

         \[\leadsto 1 - \sqrt{{x}^{-2}} \cdot \sqrt{\color{blue}{\frac{1}{81}}} \]

      7. sqrt-prod 70.1%

         \[\leadsto 1 - \color{blue}{\sqrt{{x}^{-2} \cdot \frac{1}{81}}} \]

      8. div-inv 70.1%

         \[\leadsto 1 - \sqrt{\color{blue}{\frac{{x}^{-2}}{81}}} \]

      9. sqrt-div 70.0%

         \[\leadsto 1 - \color{blue}{\frac{\sqrt{{x}^{-2}}}{\sqrt{81}}} \]

      10. sqrt-pow1 96.2%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(\frac{-2}{2}\right)}}}{\sqrt{81}} \]

      11. metadata-eval 96.2%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-1}}}{\sqrt{81}} \]

      12. inv-pow 96.2%

          \[\leadsto 1 - \frac{\color{blue}{\frac{1}{x}}}{\sqrt{81}} \]

      13. metadata-eval 96.2%

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

   7. Applied egg-rr 96.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+79} \lor \neg \left(y \leq 4 \cdot 10^{+80}\right):\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (- (* -0.3333333333333333 (* y (sqrt x))) 0.1111111111111111) x)
   (- 1.0 (/ y (sqrt (* x 9.0))))))

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 * 9.0)));
	}
	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 * 9.0d0)))
    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 * 9.0)));
	}
	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 * 9.0)))
	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(y / sqrt(Float64(x * 9.0))));
	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 * 9.0)));
	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[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. 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 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

   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 0 99.8%

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

   4. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-prod 99.9%

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

      4. pow1/2 99.9%

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

   6. Applied egg-rr 99.5%

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

      1. unpow1/2 99.9%

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

   8. Simplified 99.5%

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

4. Final simplification 98.9%

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

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. un-div-inv 99.7%

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

6. Applied egg-rr 99.7%

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

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   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. Add Preprocessing

Alternative 9: 67.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{-0.1111111111111111}{x}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1 - \frac{\frac{-0.012345679012345678}{x}}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+153}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\left(x \cdot 9\right) \cdot \left(x \cdot 9\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ -0.1111111111111111 x))))
   (if (<= y -6.5e+88)
     (/ (- 1.0 (/ (/ -0.012345679012345678 x) x)) t_0)
     (if (<= y 1.1e+153)
       (+ 1.0 (/ (/ -1.0 x) 9.0))
       (/ (+ 1.0 (/ -1.0 (* (* x 9.0) (* x 9.0)))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -6.5e+88) {
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / t_0;
	} else if (y <= 1.1e+153) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = (1.0 + (-1.0 / ((x * 9.0) * (x * 9.0)))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((-0.1111111111111111d0) / x)
    if (y <= (-6.5d+88)) then
        tmp = (1.0d0 - (((-0.012345679012345678d0) / x) / x)) / t_0
    else if (y <= 1.1d+153) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = (1.0d0 + ((-1.0d0) / ((x * 9.0d0) * (x * 9.0d0)))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -6.5e+88) {
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / t_0;
	} else if (y <= 1.1e+153) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = (1.0 + (-1.0 / ((x * 9.0) * (x * 9.0)))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (-0.1111111111111111 / x)
	tmp = 0
	if y <= -6.5e+88:
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / t_0
	elif y <= 1.1e+153:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = (1.0 + (-1.0 / ((x * 9.0) * (x * 9.0)))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(-0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -6.5e+88)
		tmp = Float64(Float64(1.0 - Float64(Float64(-0.012345679012345678 / x) / x)) / t_0);
	elseif (y <= 1.1e+153)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(Float64(x * 9.0) * Float64(x * 9.0)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (-0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= -6.5e+88)
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / t_0;
	elseif (y <= 1.1e+153)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = (1.0 + (-1.0 / ((x * 9.0) * (x * 9.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+88], N[(N[(1.0 - N[(N[(-0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 1.1e+153], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(N[(x * 9.0), $MachinePrecision] * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5000000000000002e88

   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* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.3%

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

      12. metadata-eval 99.3%

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

      13. *-commutative 99.3%

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

      14. associate-/r* 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 1.7%

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

      1. flip-+ 1.6%

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

   7. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1 - \frac{{x}^{-2}}{81}}{1 - \frac{-0.1111111111111111}{x}}} \]
   8. Step-by-step derivation

      1. div-inv 1.6%

         \[\leadsto \frac{1 - \color{blue}{{x}^{-2} \cdot \frac{1}{81}}}{1 - \frac{-0.1111111111111111}{x}} \]

      2. *-commutative 1.6%

         \[\leadsto \frac{1 - \color{blue}{\frac{1}{81} \cdot {x}^{-2}}}{1 - \frac{-0.1111111111111111}{x}} \]

      3. metadata-eval 1.6%

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

      4. metadata-eval 1.6%

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

      5. metadata-eval 1.6%

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

      6. pow-prod-up 1.6%

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

      7. inv-pow 1.6%

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

      8. inv-pow 1.6%

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

      9. swap-sqr 1.6%

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

      10. div-inv 1.6%

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

      11. associate-*l* 1.6%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 28.9%

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

      14. frac-times 28.9%

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

      15. metadata-eval 28.9%

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

      16. metadata-eval 28.9%

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

      17. frac-times 28.9%

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

      18. sqrt-prod 28.9%

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

      19. add-sqr-sqrt 28.9%

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

   9. Applied egg-rr 28.9%

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

       1. *-commutative 28.9%

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

       2. associate-*r* 28.9%

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

       3. *-commutative 28.9%

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

       4. associate-*r/ 28.9%

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

       5. *-rgt-identity 28.9%

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

       6. associate-*l/ 28.9%

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

       7. metadata-eval 28.9%

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

   11. Simplified 28.9%

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

   if -6.5000000000000002e88 < y < 1.1e153

   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* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 90.9%

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

      1. *-commutative 90.9%

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

      2. inv-pow 90.9%

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

      3. metadata-eval 90.9%

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

      4. sqrt-pow1 65.7%

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

      5. metadata-eval 65.7%

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

      6. metadata-eval 65.7%

         \[\leadsto 1 - \sqrt{{x}^{-2}} \cdot \sqrt{\color{blue}{\frac{1}{81}}} \]

      7. sqrt-prod 65.8%

         \[\leadsto 1 - \color{blue}{\sqrt{{x}^{-2} \cdot \frac{1}{81}}} \]

      8. div-inv 65.8%

         \[\leadsto 1 - \sqrt{\color{blue}{\frac{{x}^{-2}}{81}}} \]

      9. sqrt-div 65.8%

         \[\leadsto 1 - \color{blue}{\frac{\sqrt{{x}^{-2}}}{\sqrt{81}}} \]

      10. sqrt-pow1 91.0%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(\frac{-2}{2}\right)}}}{\sqrt{81}} \]

      11. metadata-eval 91.0%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-1}}}{\sqrt{81}} \]

      12. inv-pow 91.0%

          \[\leadsto 1 - \frac{\color{blue}{\frac{1}{x}}}{\sqrt{81}} \]

      13. metadata-eval 91.0%

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

   7. Applied egg-rr 91.0%

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

   if 1.1e153 < y

   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* 99.6%

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

      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* 99.6%

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

      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 3.2%

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

      1. flip-+ 20.9%

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

   7. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{\frac{1 - \frac{{x}^{-2}}{81}}{1 - \frac{-0.1111111111111111}{x}}} \]
   8. Step-by-step derivation

      1. metadata-eval 20.9%

         \[\leadsto \frac{1 - \frac{{x}^{\color{blue}{\left(-1 + -1\right)}}}{81}}{1 - \frac{-0.1111111111111111}{x}} \]

      2. pow-prod-up 20.9%

         \[\leadsto \frac{1 - \frac{\color{blue}{{x}^{-1} \cdot {x}^{-1}}}{81}}{1 - \frac{-0.1111111111111111}{x}} \]

      3. inv-pow 20.9%

         \[\leadsto \frac{1 - \frac{\color{blue}{\frac{1}{x}} \cdot {x}^{-1}}{81}}{1 - \frac{-0.1111111111111111}{x}} \]

      4. inv-pow 20.9%

         \[\leadsto \frac{1 - \frac{\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}}{81}}{1 - \frac{-0.1111111111111111}{x}} \]

      5. associate-/l* 20.9%

         \[\leadsto \frac{1 - \color{blue}{\frac{1}{x} \cdot \frac{\frac{1}{x}}{81}}}{1 - \frac{-0.1111111111111111}{x}} \]

   9. Applied egg-rr 20.9%

      \[\leadsto \frac{1 - \color{blue}{\frac{1}{x} \cdot \frac{\frac{1}{x}}{81}}}{1 - \frac{-0.1111111111111111}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 20.9%

          \[\leadsto \frac{1 - \color{blue}{\frac{\frac{1}{x} \cdot \frac{1}{x}}{81}}}{1 - \frac{-0.1111111111111111}{x}} \]

       2. metadata-eval 20.9%

          \[\leadsto \frac{1 - \frac{\frac{1}{x} \cdot \frac{1}{x}}{\color{blue}{9 \cdot 9}}}{1 - \frac{-0.1111111111111111}{x}} \]

       3. frac-times 20.9%

          \[\leadsto \frac{1 - \color{blue}{\frac{\frac{1}{x}}{9} \cdot \frac{\frac{1}{x}}{9}}}{1 - \frac{-0.1111111111111111}{x}} \]

       4. associate-/l/ 20.9%

          \[\leadsto \frac{1 - \frac{\frac{1}{x}}{9} \cdot \color{blue}{\frac{1}{9 \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       5. associate-/l/ 20.9%

          \[\leadsto \frac{1 - \color{blue}{\frac{1}{9 \cdot x}} \cdot \frac{1}{9 \cdot x}}{1 - \frac{-0.1111111111111111}{x}} \]

       6. frac-times 20.9%

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

       7. metadata-eval 20.9%

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

   11. Applied egg-rr 20.9%

       \[\leadsto \frac{1 - \color{blue}{\frac{1}{\left(9 \cdot x\right) \cdot \left(9 \cdot x\right)}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1 - \frac{\frac{-0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+153}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\left(x \cdot 9\right) \cdot \left(x \cdot 9\right)}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{-0.1111111111111111}{x}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1 - \frac{\frac{-0.012345679012345678}{x}}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+153}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{x \cdot \left(x \cdot 81\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ -0.1111111111111111 x))))
   (if (<= y -6.5e+88)
     (/ (- 1.0 (/ (/ -0.012345679012345678 x) x)) t_0)
     (if (<= y 1.1e+153)
       (+ 1.0 (/ (/ -1.0 x) 9.0))
       (/ (+ 1.0 (/ -1.0 (* x (* x 81.0)))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -6.5e+88) {
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / t_0;
	} else if (y <= 1.1e+153) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = (1.0 + (-1.0 / (x * (x * 81.0)))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((-0.1111111111111111d0) / x)
    if (y <= (-6.5d+88)) then
        tmp = (1.0d0 - (((-0.012345679012345678d0) / x) / x)) / t_0
    else if (y <= 1.1d+153) then
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    else
        tmp = (1.0d0 + ((-1.0d0) / (x * (x * 81.0d0)))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (-0.1111111111111111 / x);
	double tmp;
	if (y <= -6.5e+88) {
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / t_0;
	} else if (y <= 1.1e+153) {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	} else {
		tmp = (1.0 + (-1.0 / (x * (x * 81.0)))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (-0.1111111111111111 / x)
	tmp = 0
	if y <= -6.5e+88:
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / t_0
	elif y <= 1.1e+153:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	else:
		tmp = (1.0 + (-1.0 / (x * (x * 81.0)))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(-0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -6.5e+88)
		tmp = Float64(Float64(1.0 - Float64(Float64(-0.012345679012345678 / x) / x)) / t_0);
	elseif (y <= 1.1e+153)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(x * Float64(x * 81.0)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (-0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= -6.5e+88)
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / t_0;
	elseif (y <= 1.1e+153)
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	else
		tmp = (1.0 + (-1.0 / (x * (x * 81.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+88], N[(N[(1.0 - N[(N[(-0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 1.1e+153], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(x * N[(x * 81.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5000000000000002e88

   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* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.3%

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

      12. metadata-eval 99.3%

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

      13. *-commutative 99.3%

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

      14. associate-/r* 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 1.7%

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

      1. flip-+ 1.6%

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

   7. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1 - \frac{{x}^{-2}}{81}}{1 - \frac{-0.1111111111111111}{x}}} \]
   8. Step-by-step derivation

      1. div-inv 1.6%

         \[\leadsto \frac{1 - \color{blue}{{x}^{-2} \cdot \frac{1}{81}}}{1 - \frac{-0.1111111111111111}{x}} \]

      2. *-commutative 1.6%

         \[\leadsto \frac{1 - \color{blue}{\frac{1}{81} \cdot {x}^{-2}}}{1 - \frac{-0.1111111111111111}{x}} \]

      3. metadata-eval 1.6%

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

      4. metadata-eval 1.6%

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

      5. metadata-eval 1.6%

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

      6. pow-prod-up 1.6%

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

      7. inv-pow 1.6%

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

      8. inv-pow 1.6%

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

      9. swap-sqr 1.6%

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

      10. div-inv 1.6%

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

      11. associate-*l* 1.6%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 28.9%

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

      14. frac-times 28.9%

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

      15. metadata-eval 28.9%

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

      16. metadata-eval 28.9%

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

      17. frac-times 28.9%

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

      18. sqrt-prod 28.9%

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

      19. add-sqr-sqrt 28.9%

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

   9. Applied egg-rr 28.9%

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

       1. *-commutative 28.9%

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

       2. associate-*r* 28.9%

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

       3. *-commutative 28.9%

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

       4. associate-*r/ 28.9%

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

       5. *-rgt-identity 28.9%

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

       6. associate-*l/ 28.9%

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

       7. metadata-eval 28.9%

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

   11. Simplified 28.9%

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

   if -6.5000000000000002e88 < y < 1.1e153

   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* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 90.9%

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

      1. *-commutative 90.9%

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

      2. inv-pow 90.9%

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

      3. metadata-eval 90.9%

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

      4. sqrt-pow1 65.7%

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

      5. metadata-eval 65.7%

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

      6. metadata-eval 65.7%

         \[\leadsto 1 - \sqrt{{x}^{-2}} \cdot \sqrt{\color{blue}{\frac{1}{81}}} \]

      7. sqrt-prod 65.8%

         \[\leadsto 1 - \color{blue}{\sqrt{{x}^{-2} \cdot \frac{1}{81}}} \]

      8. div-inv 65.8%

         \[\leadsto 1 - \sqrt{\color{blue}{\frac{{x}^{-2}}{81}}} \]

      9. sqrt-div 65.8%

         \[\leadsto 1 - \color{blue}{\frac{\sqrt{{x}^{-2}}}{\sqrt{81}}} \]

      10. sqrt-pow1 91.0%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(\frac{-2}{2}\right)}}}{\sqrt{81}} \]

      11. metadata-eval 91.0%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-1}}}{\sqrt{81}} \]

      12. inv-pow 91.0%

          \[\leadsto 1 - \frac{\color{blue}{\frac{1}{x}}}{\sqrt{81}} \]

      13. metadata-eval 91.0%

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

   7. Applied egg-rr 91.0%

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

   if 1.1e153 < y

   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* 99.6%

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

      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* 99.6%

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

      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 3.2%

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

      1. flip-+ 20.9%

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

   7. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{\frac{1 - \frac{{x}^{-2}}{81}}{1 - \frac{-0.1111111111111111}{x}}} \]
   8. Step-by-step derivation

      1. metadata-eval 20.9%

         \[\leadsto \frac{1 - \frac{{x}^{\color{blue}{\left(-1 + -1\right)}}}{81}}{1 - \frac{-0.1111111111111111}{x}} \]

      2. pow-prod-up 20.9%

         \[\leadsto \frac{1 - \frac{\color{blue}{{x}^{-1} \cdot {x}^{-1}}}{81}}{1 - \frac{-0.1111111111111111}{x}} \]

      3. inv-pow 20.9%

         \[\leadsto \frac{1 - \frac{\color{blue}{\frac{1}{x}} \cdot {x}^{-1}}{81}}{1 - \frac{-0.1111111111111111}{x}} \]

      4. inv-pow 20.9%

         \[\leadsto \frac{1 - \frac{\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}}{81}}{1 - \frac{-0.1111111111111111}{x}} \]

      5. associate-/l* 20.9%

         \[\leadsto \frac{1 - \color{blue}{\frac{1}{x} \cdot \frac{\frac{1}{x}}{81}}}{1 - \frac{-0.1111111111111111}{x}} \]

   9. Applied egg-rr 20.9%

      \[\leadsto \frac{1 - \color{blue}{\frac{1}{x} \cdot \frac{\frac{1}{x}}{81}}}{1 - \frac{-0.1111111111111111}{x}} \]
   10. Step-by-step derivation

       1. *-commutative 20.9%

          \[\leadsto \frac{1 - \color{blue}{\frac{\frac{1}{x}}{81} \cdot \frac{1}{x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       2. associate-/l/ 20.9%

          \[\leadsto \frac{1 - \color{blue}{\frac{1}{81 \cdot x}} \cdot \frac{1}{x}}{1 - \frac{-0.1111111111111111}{x}} \]

       3. frac-times 20.9%

          \[\leadsto \frac{1 - \color{blue}{\frac{1 \cdot 1}{\left(81 \cdot x\right) \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]

       4. metadata-eval 20.9%

          \[\leadsto \frac{1 - \frac{\color{blue}{1}}{\left(81 \cdot x\right) \cdot x}}{1 - \frac{-0.1111111111111111}{x}} \]

   11. Applied egg-rr 20.9%

       \[\leadsto \frac{1 - \color{blue}{\frac{1}{\left(81 \cdot x\right) \cdot x}}}{1 - \frac{-0.1111111111111111}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1 - \frac{\frac{-0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+153}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{x \cdot \left(x \cdot 81\right)}}{1 - \frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1 - \frac{\frac{-0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+88)
   (/
    (- 1.0 (/ (/ -0.012345679012345678 x) x))
    (- 1.0 (/ -0.1111111111111111 x)))
   (+ 1.0 (/ (/ -1.0 x) 9.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+88) {
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.5d+88)) then
        tmp = (1.0d0 - (((-0.012345679012345678d0) / x) / x)) / (1.0d0 - ((-0.1111111111111111d0) / x))
    else
        tmp = 1.0d0 + (((-1.0d0) / x) / 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+88) {
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	} else {
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5e+88:
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x))
	else:
		tmp = 1.0 + ((-1.0 / x) / 9.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5e+88)
		tmp = Float64(Float64(1.0 - Float64(Float64(-0.012345679012345678 / x) / x)) / Float64(1.0 - Float64(-0.1111111111111111 / x)));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.5e+88)
		tmp = (1.0 - ((-0.012345679012345678 / x) / x)) / (1.0 - (-0.1111111111111111 / x));
	else
		tmp = 1.0 + ((-1.0 / x) / 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5e+88], N[(N[(1.0 - N[(N[(-0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5000000000000002e88

   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* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.3%

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

      12. metadata-eval 99.3%

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

      13. *-commutative 99.3%

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

      14. associate-/r* 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 1.7%

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

      1. flip-+ 1.6%

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

   7. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1 - \frac{{x}^{-2}}{81}}{1 - \frac{-0.1111111111111111}{x}}} \]
   8. Step-by-step derivation

      1. div-inv 1.6%

         \[\leadsto \frac{1 - \color{blue}{{x}^{-2} \cdot \frac{1}{81}}}{1 - \frac{-0.1111111111111111}{x}} \]

      2. *-commutative 1.6%

         \[\leadsto \frac{1 - \color{blue}{\frac{1}{81} \cdot {x}^{-2}}}{1 - \frac{-0.1111111111111111}{x}} \]

      3. metadata-eval 1.6%

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

      4. metadata-eval 1.6%

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

      5. metadata-eval 1.6%

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

      6. pow-prod-up 1.6%

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

      7. inv-pow 1.6%

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

      8. inv-pow 1.6%

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

      9. swap-sqr 1.6%

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

      10. div-inv 1.6%

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

      11. associate-*l* 1.6%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 28.9%

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

      14. frac-times 28.9%

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

      15. metadata-eval 28.9%

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

      16. metadata-eval 28.9%

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

      17. frac-times 28.9%

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

      18. sqrt-prod 28.9%

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

      19. add-sqr-sqrt 28.9%

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

   9. Applied egg-rr 28.9%

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

       1. *-commutative 28.9%

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

       2. associate-*r* 28.9%

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

       3. *-commutative 28.9%

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

       4. associate-*r/ 28.9%

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

       5. *-rgt-identity 28.9%

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

       6. associate-*l/ 28.9%

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

       7. metadata-eval 28.9%

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

   11. Simplified 28.9%

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

   if -6.5000000000000002e88 < y

   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* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 76.1%

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

      1. *-commutative 76.1%

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

      2. inv-pow 76.1%

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

      3. metadata-eval 76.1%

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

      4. sqrt-pow1 58.1%

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

      5. metadata-eval 58.1%

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

      6. metadata-eval 58.1%

         \[\leadsto 1 - \sqrt{{x}^{-2}} \cdot \sqrt{\color{blue}{\frac{1}{81}}} \]

      7. sqrt-prod 58.2%

         \[\leadsto 1 - \color{blue}{\sqrt{{x}^{-2} \cdot \frac{1}{81}}} \]

      8. div-inv 58.2%

         \[\leadsto 1 - \sqrt{\color{blue}{\frac{{x}^{-2}}{81}}} \]

      9. sqrt-div 58.2%

         \[\leadsto 1 - \color{blue}{\frac{\sqrt{{x}^{-2}}}{\sqrt{81}}} \]

      10. sqrt-pow1 76.2%

          \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(\frac{-2}{2}\right)}}}{\sqrt{81}} \]

      11. metadata-eval 76.2%

          \[\leadsto 1 - \frac{{x}^{\color{blue}{-1}}}{\sqrt{81}} \]

      12. inv-pow 76.2%

          \[\leadsto 1 - \frac{\color{blue}{\frac{1}{x}}}{\sqrt{81}} \]

      13. metadata-eval 76.2%

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

   7. Applied egg-rr 76.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1 - \frac{\frac{-0.012345679012345678}{x}}{x}}{1 - \frac{-0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{-1}{x}}{9}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.1% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.11d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

      1. 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* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      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 61.2%

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

   6. Taylor expanded in x around 0 60.1%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.8%

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

      15. distribute-neg-frac 99.8%

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

      16. metadata-eval 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 70.3%

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

   6. Taylor expanded in x around inf 69.9%

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

Alternative 13: 63.3% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{\frac{-1}{x}}{9} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return 1.0 + ((-1.0 / 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)
end function

Java

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

Python

def code(x, y):
	return 1.0 + ((-1.0 / x) / 9.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   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 65.3%

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

   1. *-commutative 65.3%

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

   2. inv-pow 65.3%

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

   3. metadata-eval 65.3%

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

   4. sqrt-pow1 49.9%

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

   5. metadata-eval 49.9%

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

   6. metadata-eval 49.9%

      \[\leadsto 1 - \sqrt{{x}^{-2}} \cdot \sqrt{\color{blue}{\frac{1}{81}}} \]

   7. sqrt-prod 50.0%

      \[\leadsto 1 - \color{blue}{\sqrt{{x}^{-2} \cdot \frac{1}{81}}} \]

   8. div-inv 50.0%

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{{x}^{-2}}{81}}} \]

   9. sqrt-div 50.0%

      \[\leadsto 1 - \color{blue}{\frac{\sqrt{{x}^{-2}}}{\sqrt{81}}} \]

   10. sqrt-pow1 65.4%

       \[\leadsto 1 - \frac{\color{blue}{{x}^{\left(\frac{-2}{2}\right)}}}{\sqrt{81}} \]

   11. metadata-eval 65.4%

       \[\leadsto 1 - \frac{{x}^{\color{blue}{-1}}}{\sqrt{81}} \]

   12. inv-pow 65.4%

       \[\leadsto 1 - \frac{\color{blue}{\frac{1}{x}}}{\sqrt{81}} \]

   13. metadata-eval 65.4%

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

7. Applied egg-rr 65.4%

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

8. Final simplification 65.4%

   \[\leadsto 1 + \frac{\frac{-1}{x}}{9} \]
9. Add Preprocessing

Alternative 14: 63.3% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   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* 99.6%

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

   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 65.4%

   \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111}{x}} \]
6. Add Preprocessing

Alternative 15: 33.0% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. 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* 99.6%

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

   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* 99.6%

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

   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 65.4%

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

6. Taylor expanded in x around inf 33.0%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer Target 1: 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 2024137 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x)))))

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