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

Percentage Accurate: 99.7% → 99.7%

Time: 9.6s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+56} \lor \neg \left(y \leq 2.4 \cdot 10^{+60}\right):\\ \;\;\;\;1 - y \cdot \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.5e+56) (not (<= y 2.4e+60)))
   (- 1.0 (* y (/ 0.3333333333333333 (sqrt x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.5e+56) || !(y <= 2.4e+60)) {
		tmp = 1.0 - (y * (0.3333333333333333 / sqrt(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 <= (-8.5d+56)) .or. (.not. (y <= 2.4d+60))) then
        tmp = 1.0d0 - (y * (0.3333333333333333d0 / sqrt(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 <= -8.5e+56) || !(y <= 2.4e+60)) {
		tmp = 1.0 - (y * (0.3333333333333333 / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.5e+56) or not (y <= 2.4e+60):
		tmp = 1.0 - (y * (0.3333333333333333 / math.sqrt(x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.5e+56) || !(y <= 2.4e+60))
		tmp = Float64(1.0 - Float64(y * Float64(0.3333333333333333 / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8.5e+56) || ~((y <= 2.4e+60)))
		tmp = 1.0 - (y * (0.3333333333333333 / sqrt(x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.5e+56], N[Not[LessEqual[y, 2.4e+60]], $MachinePrecision]], N[(1.0 - N[(y * N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.4999999999999998e56 or 2.4e60 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 94.4%

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

      1. metadata-eval 94.4%

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

      2. *-commutative 94.4%

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

      3. sqrt-div 94.4%

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

      4. metadata-eval 94.4%

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

      5. div-inv 94.6%

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

      6. times-frac 94.6%

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

      7. *-un-lft-identity 94.6%

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

      8. clear-num 94.5%

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

      9. *-un-lft-identity 94.5%

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

      10. times-frac 94.6%

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

      11. metadata-eval 94.6%

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

   5. Applied egg-rr 94.6%

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

      1. associate-/r* 94.5%

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

      2. metadata-eval 94.5%

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

      3. associate-/r/ 94.5%

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

      4. *-commutative 94.5%

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

   7. Simplified 94.5%

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

   if -8.4999999999999998e56 < y < 2.4e60

   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* 99.6%

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

      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 97.8%

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

      1. metadata-eval 97.8%

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

      2. distribute-neg-frac 97.8%

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

      3. add-sqr-sqrt 97.7%

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

      4. sqrt-unprod 72.4%

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

      5. frac-times 72.5%

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

      6. metadata-eval 72.5%

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

      7. metadata-eval 72.5%

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

      8. frac-times 72.4%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 48.0%

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

   7. Applied egg-rr 48.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 72.4%

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

      3. frac-times 72.5%

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

      4. metadata-eval 72.5%

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

      5. pow2 72.5%

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

   9. Applied egg-rr 72.5%

      \[\leadsto 1 + \left(-\color{blue}{\sqrt{\frac{0.012345679012345678}{{x}^{2}}}}\right) \]
   10. Step-by-step derivation

       1. clear-num 72.5%

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

       2. sqrt-div 73.0%

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

       3. metadata-eval 73.0%

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

       4. div-inv 73.1%

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

       5. unpow2 73.1%

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

       6. metadata-eval 73.1%

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

       7. metadata-eval 73.1%

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

       8. swap-sqr 73.2%

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

       9. sqrt-unprod 97.7%

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

       10. add-sqr-sqrt 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+56} \lor \neg \left(y \leq 2.4 \cdot 10^{+60}\right):\\ \;\;\;\;1 - y \cdot \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+54} \lor \neg \left(y \leq 1.4 \cdot 10^{+60}\right):\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.9e+54) (not (<= y 1.4e+60)))
   (- 1.0 (/ 0.3333333333333333 (/ (sqrt x) y)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.9e+54) || !(y <= 1.4e+60)) {
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	} 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 <= (-4.9d+54)) .or. (.not. (y <= 1.4d+60))) then
        tmp = 1.0d0 - (0.3333333333333333d0 / (sqrt(x) / y))
    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 <= -4.9e+54) || !(y <= 1.4e+60)) {
		tmp = 1.0 - (0.3333333333333333 / (Math.sqrt(x) / y));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.9e+54) or not (y <= 1.4e+60):
		tmp = 1.0 - (0.3333333333333333 / (math.sqrt(x) / y))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.9e+54) || !(y <= 1.4e+60))
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(sqrt(x) / y)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.9e+54) || ~((y <= 1.4e+60)))
		tmp = 1.0 - (0.3333333333333333 / (sqrt(x) / y));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.9e+54], N[Not[LessEqual[y, 1.4e+60]], $MachinePrecision]], N[(1.0 - N[(0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.90000000000000001e54 or 1.4e60 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 94.4%

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

      1. *-commutative 94.4%

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

      2. sqrt-div 94.4%

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

      3. metadata-eval 94.4%

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

      4. div-inv 94.6%

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

      5. clear-num 94.5%

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

      6. add-sqr-sqrt 48.1%

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

      7. sqrt-unprod 29.1%

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

      8. sqr-neg 29.1%

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

      9. sqrt-unprod 3.0%

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

      10. add-sqr-sqrt 6.5%

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

      11. distribute-neg-frac2 6.5%

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

      12. div-inv 6.5%

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

      13. distribute-neg-frac2 6.5%

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

      14. add-sqr-sqrt 3.0%

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

      15. sqrt-unprod 29.1%

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

      16. sqr-neg 29.1%

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

      17. sqrt-unprod 48.1%

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

      18. add-sqr-sqrt 94.5%

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

   5. Applied egg-rr 94.5%

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

   if -4.90000000000000001e54 < y < 1.4e60

   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* 99.6%

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

      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 97.8%

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

      1. metadata-eval 97.8%

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

      2. distribute-neg-frac 97.8%

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

      3. add-sqr-sqrt 97.7%

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

      4. sqrt-unprod 72.4%

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

      5. frac-times 72.5%

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

      6. metadata-eval 72.5%

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

      7. metadata-eval 72.5%

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

      8. frac-times 72.4%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 48.0%

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

   7. Applied egg-rr 48.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 72.4%

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

      3. frac-times 72.5%

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

      4. metadata-eval 72.5%

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

      5. pow2 72.5%

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

   9. Applied egg-rr 72.5%

      \[\leadsto 1 + \left(-\color{blue}{\sqrt{\frac{0.012345679012345678}{{x}^{2}}}}\right) \]
   10. Step-by-step derivation

       1. clear-num 72.5%

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

       2. sqrt-div 73.0%

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

       3. metadata-eval 73.0%

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

       4. div-inv 73.1%

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

       5. unpow2 73.1%

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

       6. metadata-eval 73.1%

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

       7. metadata-eval 73.1%

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

       8. swap-sqr 73.2%

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

       9. sqrt-unprod 97.7%

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

       10. add-sqr-sqrt 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+54} \lor \neg \left(y \leq 1.4 \cdot 10^{+60}\right):\\ \;\;\;\;1 - \frac{0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{+55} \lor \neg \left(y \leq 1.18 \cdot 10^{+60}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.08e+55) (not (<= y 1.18e+60)))
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.08e+55) || !(y <= 1.18e+60)) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} 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 <= (-1.08d+55)) .or. (.not. (y <= 1.18d+60))) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    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 <= -1.08e+55) || !(y <= 1.18e+60)) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.08e+55) or not (y <= 1.18e+60):
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.08e+55) || !(y <= 1.18e+60))
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.08e+55) || ~((y <= 1.18e+60)))
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.08e+55], N[Not[LessEqual[y, 1.18e+60]], $MachinePrecision]], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.08000000000000004e55 or 1.18000000000000008e60 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 94.4%

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

      1. metadata-eval 94.4%

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

      2. *-commutative 94.4%

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

      3. sqrt-div 94.4%

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

      4. metadata-eval 94.4%

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

      5. div-inv 94.6%

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

      6. times-frac 94.6%

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

      7. *-un-lft-identity 94.6%

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

   5. Applied egg-rr 94.6%

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

      1. *-commutative 94.6%

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

      2. metadata-eval 94.6%

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

      3. sqrt-prod 94.7%

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

      4. pow1/2 94.7%

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

   7. Applied egg-rr 94.7%

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

      1. unpow1/2 94.7%

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

   9. Simplified 94.7%

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

   if -1.08000000000000004e55 < y < 1.18000000000000008e60

   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* 99.6%

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

      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 97.8%

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

      1. metadata-eval 97.8%

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

      2. distribute-neg-frac 97.8%

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

      3. add-sqr-sqrt 97.7%

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

      4. sqrt-unprod 72.4%

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

      5. frac-times 72.5%

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

      6. metadata-eval 72.5%

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

      7. metadata-eval 72.5%

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

      8. frac-times 72.4%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 48.0%

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

   7. Applied egg-rr 48.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 72.4%

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

      3. frac-times 72.5%

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

      4. metadata-eval 72.5%

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

      5. pow2 72.5%

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

   9. Applied egg-rr 72.5%

      \[\leadsto 1 + \left(-\color{blue}{\sqrt{\frac{0.012345679012345678}{{x}^{2}}}}\right) \]
   10. Step-by-step derivation

       1. clear-num 72.5%

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

       2. sqrt-div 73.0%

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

       3. metadata-eval 73.0%

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

       4. div-inv 73.1%

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

       5. unpow2 73.1%

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

       6. metadata-eval 73.1%

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

       7. metadata-eval 73.1%

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

       8. swap-sqr 73.2%

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

       9. sqrt-unprod 97.7%

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

       10. add-sqr-sqrt 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{+55} \lor \neg \left(y \leq 1.18 \cdot 10^{+60}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+67} \lor \neg \left(y \leq 9.2 \cdot 10^{+79}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.15e+67) (not (<= y 9.2e+79)))
   (* y (/ -0.3333333333333333 (sqrt x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.15e+67) || !(y <= 9.2e+79)) {
		tmp = y * (-0.3333333333333333 / sqrt(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 <= (-1.15d+67)) .or. (.not. (y <= 9.2d+79))) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(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 <= -1.15e+67) || !(y <= 9.2e+79)) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.15e+67) or not (y <= 9.2e+79):
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.15e+67) || !(y <= 9.2e+79))
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.15e+67) || ~((y <= 9.2e+79)))
		tmp = y * (-0.3333333333333333 / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.15e+67], N[Not[LessEqual[y, 9.2e+79]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1499999999999999e67 or 9.2000000000000002e79 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 93.0%

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

      1. *-commutative 93.0%

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

      2. *-commutative 93.0%

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

      3. associate-*r* 93.1%

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

   6. Simplified 93.1%

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

      1. *-commutative 93.1%

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

      2. sqrt-div 93.0%

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

      3. metadata-eval 93.0%

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

      4. un-div-inv 93.1%

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

   8. Applied egg-rr 93.1%

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

   if -1.1499999999999999e67 < y < 9.2000000000000002e79

   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* 99.6%

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

      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 95.2%

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

      1. metadata-eval 95.2%

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

      2. distribute-neg-frac 95.2%

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

      3. add-sqr-sqrt 95.1%

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

      4. sqrt-unprod 70.1%

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

      5. frac-times 70.1%

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

      6. metadata-eval 70.1%

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

      7. metadata-eval 70.1%

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

      8. frac-times 70.1%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 47.4%

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

   7. Applied egg-rr 47.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 70.1%

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

      3. frac-times 70.1%

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

      4. metadata-eval 70.1%

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

      5. pow2 70.1%

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

   9. Applied egg-rr 70.1%

      \[\leadsto 1 + \left(-\color{blue}{\sqrt{\frac{0.012345679012345678}{{x}^{2}}}}\right) \]
   10. Step-by-step derivation

       1. clear-num 70.1%

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

       2. sqrt-div 70.6%

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

       3. metadata-eval 70.6%

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

       4. div-inv 70.7%

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

       5. unpow2 70.7%

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

       6. metadata-eval 70.7%

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

       7. metadata-eval 70.7%

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

       8. swap-sqr 70.7%

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

       9. sqrt-unprod 95.2%

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

       10. add-sqr-sqrt 95.4%

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

   11. Applied egg-rr 95.4%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+67} \lor \neg \left(y \leq 9.2 \cdot 10^{+79}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+68}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+79}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3e+68)
   (* -0.3333333333333333 (* y (pow x -0.5)))
   (if (<= y 2.6e+79)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* y (/ -0.3333333333333333 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3e+68) {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	} else if (y <= 2.6e+79) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3d+68)) then
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    else if (y <= 2.6d+79) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3e+68) {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	} else if (y <= 2.6e+79) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3e+68:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	elif y <= 2.6e+79:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3e+68)
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	elseif (y <= 2.6e+79)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3e+68)
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	elseif (y <= 2.6e+79)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = y * (-0.3333333333333333 / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3e+68], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+79], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0000000000000002e68

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in y around inf 91.9%

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

      1. *-commutative 91.9%

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

      2. *-commutative 91.9%

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

      3. associate-*r* 91.9%

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

   6. Simplified 91.9%

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

      1. *-un-lft-identity 91.9%

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

      2. inv-pow 91.9%

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

      3. sqrt-pow1 91.9%

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

      4. metadata-eval 91.9%

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

   8. Applied egg-rr 91.9%

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

      1. *-lft-identity 91.9%

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

   10. Simplified 91.9%

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

       1. add-sqr-sqrt 91.5%

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

       2. sqrt-unprod 59.3%

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

       3. associate-*r* 59.2%

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

       4. associate-*r* 59.3%

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

       5. swap-sqr 59.3%

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

       6. *-commutative 59.3%

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

       7. *-commutative 59.3%

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

       8. metadata-eval 59.3%

          \[\leadsto \sqrt{\left(\left({x}^{-0.5} \cdot y\right) \cdot \left({x}^{-0.5} \cdot y\right)\right) \cdot \color{blue}{0.1111111111111111}} \]

       9. metadata-eval 59.3%

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

       10. swap-sqr 59.3%

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

       11. associate-*r* 59.2%

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

       12. associate-*r* 59.2%

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

       13. sqrt-unprod 0.0%

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

       14. add-log-exp 0.0%

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

       15. add-sqr-sqrt 0.5%

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

       16. *-un-lft-identity 0.5%

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

   12. Applied egg-rr 91.9%

       \[\leadsto \color{blue}{0 + y \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 91.9%

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

       2. associate-*r* 91.9%

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

       3. *-commutative 91.9%

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

   14. Simplified 91.9%

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

   if -3.0000000000000002e68 < y < 2.60000000000000015e79

   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* 99.6%

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

      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 95.2%

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

      1. metadata-eval 95.2%

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

      2. distribute-neg-frac 95.2%

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

      3. add-sqr-sqrt 95.1%

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

      4. sqrt-unprod 70.1%

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

      5. frac-times 70.1%

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

      6. metadata-eval 70.1%

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

      7. metadata-eval 70.1%

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

      8. frac-times 70.1%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 47.4%

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

   7. Applied egg-rr 47.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 70.1%

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

      3. frac-times 70.1%

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

      4. metadata-eval 70.1%

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

      5. pow2 70.1%

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

   9. Applied egg-rr 70.1%

      \[\leadsto 1 + \left(-\color{blue}{\sqrt{\frac{0.012345679012345678}{{x}^{2}}}}\right) \]
   10. Step-by-step derivation

       1. clear-num 70.1%

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

       2. sqrt-div 70.6%

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

       3. metadata-eval 70.6%

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

       4. div-inv 70.7%

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

       5. unpow2 70.7%

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

       6. metadata-eval 70.7%

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

       7. metadata-eval 70.7%

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

       8. swap-sqr 70.7%

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

       9. sqrt-unprod 95.2%

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

       10. add-sqr-sqrt 95.4%

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

   11. Applied egg-rr 95.4%

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

   if 2.60000000000000015e79 < 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 y around inf 94.2%

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

      1. *-commutative 94.2%

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

      2. *-commutative 94.2%

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

      3. associate-*r* 94.3%

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

   6. Simplified 94.3%

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

      1. *-commutative 94.3%

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

      2. sqrt-div 94.2%

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

      3. metadata-eval 94.2%

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

      4. un-div-inv 94.3%

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

   8. Applied egg-rr 94.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+68}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+79}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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* 99.4%

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

      4. metadata-eval 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. neg-mul-1 99.4%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.3%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in x around 0 98.3%

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

   if 0.110000000000000001 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. metadata-eval 98.9%

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

      2. *-commutative 98.9%

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

      3. sqrt-div 98.9%

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

      4. metadata-eval 98.9%

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

      5. div-inv 99.0%

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

      6. times-frac 99.0%

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

      7. *-un-lft-identity 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. metadata-eval 99.0%

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

      3. sqrt-prod 99.1%

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

      4. pow1/2 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. unpow1/2 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 98.7%

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

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 + (0.1111111111111111 * (-1.0 / x))) - (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 + (0.1111111111111111d0 * ((-1.0d0) / x))) - (y / (3.0d0 * sqrt(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. 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. Step-by-step derivation

   1. div-inv 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 10: 62.4% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 60.4%

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

6. Final simplification 60.4%

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

Alternative 11: 62.4% accurate, 16.1× speedup

Math

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

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(-1.0 / Float64(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[(-1.0 / N[(x * 9.0), $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. 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* 99.5%

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

   15. distribute-neg-frac 99.5%

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

   16. metadata-eval 99.5%

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

   17. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in y around 0 60.3%

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

   1. metadata-eval 60.3%

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

   2. distribute-neg-frac 60.3%

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

   3. add-sqr-sqrt 60.2%

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

   4. sqrt-unprod 46.4%

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

   5. frac-times 46.5%

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

   6. metadata-eval 46.5%

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

   7. metadata-eval 46.5%

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

   8. frac-times 46.4%

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

   9. sqrt-unprod 0.0%

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

   10. add-sqr-sqrt 30.6%

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

7. Applied egg-rr 30.6%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 46.4%

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

   3. frac-times 46.5%

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

   4. metadata-eval 46.5%

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

   5. pow2 46.5%

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

9. Applied egg-rr 46.5%

   \[\leadsto 1 + \left(-\color{blue}{\sqrt{\frac{0.012345679012345678}{{x}^{2}}}}\right) \]
10. Step-by-step derivation

    1. clear-num 46.5%

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

    2. sqrt-div 46.8%

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

    3. metadata-eval 46.8%

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

    4. div-inv 46.8%

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

    5. unpow2 46.8%

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

    6. metadata-eval 46.8%

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

    7. metadata-eval 46.8%

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

    8. swap-sqr 46.8%

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

    9. sqrt-unprod 60.3%

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

    10. add-sqr-sqrt 60.4%

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

11. Applied egg-rr 60.4%

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

12. Final simplification 60.4%

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

Alternative 12: 32.6% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.5%

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

   2. un-div-inv 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 64.1%

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

8. Taylor expanded in y around 0 31.1%

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

   1. clear-num 31.2%

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

   2. associate-/r/ 31.2%

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

10. Applied egg-rr 31.2%

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

11. Final simplification 31.2%

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

Alternative 13: 62.4% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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* 99.5%

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

   15. distribute-neg-frac 99.5%

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

   16. metadata-eval 99.5%

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

   17. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in y around 0 60.3%

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

6. Final simplification 60.3%

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

Alternative 14: 32.6% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.5%

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

   2. un-div-inv 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 64.1%

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

8. Taylor expanded in y around 0 31.1%

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

9. Final simplification 31.1%

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

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024075 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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