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

Percentage Accurate: 99.7% → 99.7%

Time: 19.9s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

4. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+84} \lor \neg \left(y \leq 5.2 \cdot 10^{+44}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.9e+84) (not (<= y 5.2e+44)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e+84) || !(y <= 5.2e+44)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.9d+84)) .or. (.not. (y <= 5.2d+44))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e+84) || !(y <= 5.2e+44)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.9e+84) or not (y <= 5.2e+44):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.9e+84) || !(y <= 5.2e+44))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.9e+84) || ~((y <= 5.2e+44)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.9e+84], N[Not[LessEqual[y, 5.2e+44]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.89999999999999989e84 or 5.1999999999999998e44 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.4%

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

      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. Taylor expanded in x around inf 95.1%

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

   if -2.89999999999999989e84 < y < 5.1999999999999998e44

   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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.2%

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

      1. div-inv 94.2%

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

      2. metadata-eval 94.2%

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

      3. cancel-sign-sub-inv 94.2%

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

      4. *-commutative 94.2%

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

      5. metadata-eval 94.2%

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

      6. div-inv 94.3%

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

      7. associate-/r* 94.3%

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

      8. add-sqr-sqrt 94.0%

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

      9. sqrt-unprod 66.3%

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

      10. *-commutative 66.3%

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

      11. associate-/r* 66.3%

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

      12. metadata-eval 66.3%

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

      13. *-commutative 66.3%

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

      14. associate-/r* 66.3%

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

      15. metadata-eval 66.3%

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

      16. frac-times 66.3%

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

      17. metadata-eval 66.3%

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

      18. metadata-eval 66.3%

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

      19. frac-times 66.3%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 40.4%

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

   7. Applied egg-rr 40.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 66.3%

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

      3. frac-times 66.3%

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

      4. metadata-eval 66.3%

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

      5. metadata-eval 66.3%

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

      6. frac-times 66.3%

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

      7. clear-num 66.3%

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

      8. clear-num 66.3%

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

      9. frac-times 66.3%

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

      10. div-inv 66.3%

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

      11. metadata-eval 66.3%

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

      12. div-inv 66.3%

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

      13. metadata-eval 66.3%

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

      14. frac-times 66.3%

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

      15. sqrt-unprod 94.0%

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

      16. add-sqr-sqrt 94.3%

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

      17. inv-pow 94.3%

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

   9. Applied egg-rr 94.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+84} \lor \neg \left(y \leq 5.2 \cdot 10^{+44}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+84} \lor \neg \left(y \leq 1.9 \cdot 10^{+41}\right):\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.9e+84) (not (<= y 1.9e+41)))
   (+ 1.0 (/ (/ y -3.0) (sqrt x)))
   (- 1.0 (pow (* x 9.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e+84) || !(y <= 1.9e+41)) {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	} else {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.9d+84)) .or. (.not. (y <= 1.9d+41))) then
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    else
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e+84) || !(y <= 1.9e+41)) {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	} else {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.9e+84) or not (y <= 1.9e+41):
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	else:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.9e+84) || !(y <= 1.9e+41))
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	else
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.9e+84) || ~((y <= 1.9e+41)))
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	else
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.9e+84], N[Not[LessEqual[y, 1.9e+41]], $MachinePrecision]], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.89999999999999989e84 or 1.9000000000000001e41 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 95.0%

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

      1. associate-*r* 95.1%

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

      2. *-commutative 95.1%

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

      3. associate-*l* 95.0%

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

   7. Simplified 95.0%

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

      1. associate-*r* 95.1%

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

      2. sqrt-div 95.1%

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

      3. metadata-eval 95.1%

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

      4. associate-*l/ 95.1%

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

      5. metadata-eval 95.1%

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

      6. metadata-eval 95.1%

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

      7. associate-/r* 95.1%

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

      8. neg-mul-1 95.1%

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

      9. un-div-inv 95.1%

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

   9. Applied egg-rr 95.3%

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

       1. frac-2neg 95.3%

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

       2. div-inv 95.2%

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

       3. remove-double-neg 95.2%

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

       4. distribute-rgt-neg-in 95.2%

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

       5. metadata-eval 95.2%

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

   11. Applied egg-rr 95.2%

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

       1. expm1-log1p-u 44.3%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{1}{\sqrt{x} \cdot -3}\right)\right)} \]

       2. expm1-undefine 44.3%

          \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \frac{1}{\sqrt{x} \cdot -3}\right)} - 1\right)} \]

       3. *-commutative 44.3%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{-3 \cdot \sqrt{x}}}\right)} - 1\right) \]

       4. associate-/r* 44.3%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{\frac{1}{-3}}{\sqrt{x}}}\right)} - 1\right) \]

       5. metadata-eval 44.3%

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

   13. Applied egg-rr 44.3%

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

   15. Simplified 95.2%

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

   if -2.89999999999999989e84 < y < 1.9000000000000001e41

   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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.2%

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

      1. div-inv 94.2%

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

      2. metadata-eval 94.2%

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

      3. cancel-sign-sub-inv 94.2%

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

      4. *-commutative 94.2%

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

      5. metadata-eval 94.2%

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

      6. div-inv 94.3%

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

      7. associate-/r* 94.3%

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

      8. add-sqr-sqrt 94.0%

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

      9. sqrt-unprod 66.3%

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

      10. *-commutative 66.3%

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

      11. associate-/r* 66.3%

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

      12. metadata-eval 66.3%

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

      13. *-commutative 66.3%

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

      14. associate-/r* 66.3%

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

      15. metadata-eval 66.3%

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

      16. frac-times 66.3%

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

      17. metadata-eval 66.3%

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

      18. metadata-eval 66.3%

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

      19. frac-times 66.3%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 40.4%

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

   7. Applied egg-rr 40.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 66.3%

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

      3. frac-times 66.3%

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

      4. metadata-eval 66.3%

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

      5. metadata-eval 66.3%

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

      6. frac-times 66.3%

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

      7. clear-num 66.3%

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

      8. clear-num 66.3%

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

      9. frac-times 66.3%

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

      10. div-inv 66.3%

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

      11. metadata-eval 66.3%

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

      12. div-inv 66.3%

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

      13. metadata-eval 66.3%

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

      14. frac-times 66.3%

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

      15. sqrt-unprod 94.0%

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

      16. add-sqr-sqrt 94.3%

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

      17. inv-pow 94.3%

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

   9. Applied egg-rr 94.3%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+84} \lor \neg \left(y \leq 1.9 \cdot 10^{+41}\right):\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+84}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+44}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e+84)
   (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
   (if (<= y 2.2e+44)
     (- 1.0 (pow (* x 9.0) -1.0))
     (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+84) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= 2.2e+44) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.2d+84)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= 2.2d+44) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+84) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= 2.2e+44) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e+84:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= 2.2e+44:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e+84)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= 2.2e+44)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.2e+84)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= 2.2e+44)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e+84], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+44], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2000000000000001e84

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      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. Taylor expanded in x around inf 99.4%

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

      1. clear-num 99.3%

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

      2. un-div-inv 99.5%

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

   7. Applied egg-rr 99.5%

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

   if -3.2000000000000001e84 < y < 2.19999999999999996e44

   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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.2%

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

      1. div-inv 94.2%

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

      2. metadata-eval 94.2%

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

      3. cancel-sign-sub-inv 94.2%

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

      4. *-commutative 94.2%

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

      5. metadata-eval 94.2%

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

      6. div-inv 94.3%

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

      7. associate-/r* 94.3%

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

      8. add-sqr-sqrt 94.0%

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

      9. sqrt-unprod 66.3%

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

      10. *-commutative 66.3%

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

      11. associate-/r* 66.3%

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

      12. metadata-eval 66.3%

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

      13. *-commutative 66.3%

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

      14. associate-/r* 66.3%

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

      15. metadata-eval 66.3%

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

      16. frac-times 66.3%

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

      17. metadata-eval 66.3%

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

      18. metadata-eval 66.3%

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

      19. frac-times 66.3%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 40.4%

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

   7. Applied egg-rr 40.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 66.3%

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

      3. frac-times 66.3%

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

      4. metadata-eval 66.3%

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

      5. metadata-eval 66.3%

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

      6. frac-times 66.3%

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

      7. clear-num 66.3%

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

      8. clear-num 66.3%

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

      9. frac-times 66.3%

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

      10. div-inv 66.3%

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

      11. metadata-eval 66.3%

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

      12. div-inv 66.3%

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

      13. metadata-eval 66.3%

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

      14. frac-times 66.3%

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

      15. sqrt-unprod 94.0%

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

      16. add-sqr-sqrt 94.3%

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

      17. inv-pow 94.3%

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

   9. Applied egg-rr 94.3%

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

   if 2.19999999999999996e44 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.4%

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

      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. Taylor expanded in x around inf 91.6%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+84}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+44}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+84}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+45}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.9e+84)
   (+ 1.0 (/ (/ y (sqrt x)) -3.0))
   (if (<= y 9.6e+45)
     (- 1.0 (pow (* x 9.0) -1.0))
     (+ 1.0 (/ (/ y -3.0) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+84) {
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	} else if (y <= 9.6e+45) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.9d+84)) then
        tmp = 1.0d0 + ((y / sqrt(x)) / (-3.0d0))
    else if (y <= 9.6d+45) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+84) {
		tmp = 1.0 + ((y / Math.sqrt(x)) / -3.0);
	} else if (y <= 9.6e+45) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.9e+84:
		tmp = 1.0 + ((y / math.sqrt(x)) / -3.0)
	elif y <= 9.6e+45:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.9e+84)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) / -3.0));
	elseif (y <= 9.6e+45)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.9e+84)
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	elseif (y <= 9.6e+45)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.9e+84], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.6e+45], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.89999999999999989e84

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.3%

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

   7. Simplified 99.3%

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

      1. associate-*r* 99.5%

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

      2. sqrt-div 99.5%

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

      3. metadata-eval 99.5%

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

      4. associate-*l/ 99.5%

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

      5. metadata-eval 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-/r* 99.5%

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

      8. neg-mul-1 99.5%

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

      9. un-div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. frac-2neg 99.5%

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

       2. div-inv 99.5%

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

       3. remove-double-neg 99.5%

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

       4. distribute-rgt-neg-in 99.5%

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

       5. metadata-eval 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. un-div-inv 99.5%

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

       2. associate-/r* 99.7%

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

   13. Applied egg-rr 99.7%

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

   if -2.89999999999999989e84 < y < 9.59999999999999958e45

   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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.2%

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

      1. div-inv 94.2%

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

      2. metadata-eval 94.2%

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

      3. cancel-sign-sub-inv 94.2%

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

      4. *-commutative 94.2%

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

      5. metadata-eval 94.2%

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

      6. div-inv 94.3%

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

      7. associate-/r* 94.3%

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

      8. add-sqr-sqrt 94.0%

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

      9. sqrt-unprod 66.3%

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

      10. *-commutative 66.3%

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

      11. associate-/r* 66.3%

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

      12. metadata-eval 66.3%

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

      13. *-commutative 66.3%

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

      14. associate-/r* 66.3%

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

      15. metadata-eval 66.3%

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

      16. frac-times 66.3%

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

      17. metadata-eval 66.3%

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

      18. metadata-eval 66.3%

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

      19. frac-times 66.3%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 40.4%

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

   7. Applied egg-rr 40.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 66.3%

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

      3. frac-times 66.3%

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

      4. metadata-eval 66.3%

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

      5. metadata-eval 66.3%

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

      6. frac-times 66.3%

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

      7. clear-num 66.3%

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

      8. clear-num 66.3%

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

      9. frac-times 66.3%

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

      10. div-inv 66.3%

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

      11. metadata-eval 66.3%

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

      12. div-inv 66.3%

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

      13. metadata-eval 66.3%

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

      14. frac-times 66.3%

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

      15. sqrt-unprod 94.0%

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

      16. add-sqr-sqrt 94.3%

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

      17. inv-pow 94.3%

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

   9. Applied egg-rr 94.3%

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

   if 9.59999999999999958e45 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. sub-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.3%

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

      12. metadata-eval 99.3%

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

      13. *-commutative 99.3%

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

      14. associate-/r* 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 91.5%

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

      1. associate-*r* 91.5%

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

      2. *-commutative 91.5%

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

      3. associate-*l* 91.5%

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

   7. Simplified 91.5%

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

      1. associate-*r* 91.5%

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

      2. sqrt-div 91.5%

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

      3. metadata-eval 91.5%

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

      4. associate-*l/ 91.5%

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

      5. metadata-eval 91.5%

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

      6. metadata-eval 91.5%

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

      7. associate-/r* 91.5%

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

      8. neg-mul-1 91.5%

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

      9. un-div-inv 91.5%

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

   9. Applied egg-rr 91.7%

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

       1. frac-2neg 91.7%

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

       2. div-inv 91.6%

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

       3. remove-double-neg 91.6%

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

       4. distribute-rgt-neg-in 91.6%

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

       5. metadata-eval 91.6%

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

   11. Applied egg-rr 91.6%

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

       1. expm1-log1p-u 5.2%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{1}{\sqrt{x} \cdot -3}\right)\right)} \]

       2. expm1-undefine 5.2%

          \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \frac{1}{\sqrt{x} \cdot -3}\right)} - 1\right)} \]

       3. *-commutative 5.2%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{-3 \cdot \sqrt{x}}}\right)} - 1\right) \]

       4. associate-/r* 5.2%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{\frac{1}{-3}}{\sqrt{x}}}\right)} - 1\right) \]

       5. metadata-eval 5.2%

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

   13. Applied egg-rr 5.2%

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

   15. Simplified 91.7%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+84}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+45}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+84}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+48}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.9e+84)
   (+ 1.0 (/ (/ y (sqrt x)) -3.0))
   (if (<= y 1.5e+48)
     (- 1.0 (pow (* x 9.0) -1.0))
     (- 1.0 (/ y (sqrt (* x 9.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+84) {
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	} else if (y <= 1.5e+48) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} 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 (y <= (-2.9d+84)) then
        tmp = 1.0d0 + ((y / sqrt(x)) / (-3.0d0))
    else if (y <= 1.5d+48) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    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 (y <= -2.9e+84) {
		tmp = 1.0 + ((y / Math.sqrt(x)) / -3.0);
	} else if (y <= 1.5e+48) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.9e+84:
		tmp = 1.0 + ((y / math.sqrt(x)) / -3.0)
	elif y <= 1.5e+48:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.9e+84)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) / -3.0));
	elseif (y <= 1.5e+48)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	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 (y <= -2.9e+84)
		tmp = 1.0 + ((y / sqrt(x)) / -3.0);
	elseif (y <= 1.5e+48)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.9e+84], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+48], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.89999999999999989e84

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-/r* 99.5%

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

      15. distribute-neg-frac 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.3%

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

   7. Simplified 99.3%

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

      1. associate-*r* 99.5%

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

      2. sqrt-div 99.5%

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

      3. metadata-eval 99.5%

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

      4. associate-*l/ 99.5%

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

      5. metadata-eval 99.5%

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

      6. metadata-eval 99.5%

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

      7. associate-/r* 99.5%

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

      8. neg-mul-1 99.5%

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

      9. un-div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. frac-2neg 99.5%

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

       2. div-inv 99.5%

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

       3. remove-double-neg 99.5%

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

       4. distribute-rgt-neg-in 99.5%

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

       5. metadata-eval 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. un-div-inv 99.5%

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

       2. associate-/r* 99.7%

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

   13. Applied egg-rr 99.7%

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

   if -2.89999999999999989e84 < y < 1.5e48

   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* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.2%

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

      1. div-inv 94.2%

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

      2. metadata-eval 94.2%

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

      3. cancel-sign-sub-inv 94.2%

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

      4. *-commutative 94.2%

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

      5. metadata-eval 94.2%

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

      6. div-inv 94.3%

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

      7. associate-/r* 94.3%

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

      8. add-sqr-sqrt 94.0%

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

      9. sqrt-unprod 66.3%

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

      10. *-commutative 66.3%

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

      11. associate-/r* 66.3%

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

      12. metadata-eval 66.3%

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

      13. *-commutative 66.3%

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

      14. associate-/r* 66.3%

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

      15. metadata-eval 66.3%

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

      16. frac-times 66.3%

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

      17. metadata-eval 66.3%

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

      18. metadata-eval 66.3%

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

      19. frac-times 66.3%

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

      20. sqrt-unprod 0.0%

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

      21. add-sqr-sqrt 40.4%

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

   7. Applied egg-rr 40.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 66.3%

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

      3. frac-times 66.3%

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

      4. metadata-eval 66.3%

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

      5. metadata-eval 66.3%

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

      6. frac-times 66.3%

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

      7. clear-num 66.3%

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

      8. clear-num 66.3%

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

      9. frac-times 66.3%

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

      10. div-inv 66.3%

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

      11. metadata-eval 66.3%

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

      12. div-inv 66.3%

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

      13. metadata-eval 66.3%

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

      14. frac-times 66.3%

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

      15. sqrt-unprod 94.0%

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

      16. add-sqr-sqrt 94.3%

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

      17. inv-pow 94.3%

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

   9. Applied egg-rr 94.3%

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

   if 1.5e48 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 91.8%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+84}:\\ \;\;\;\;1 + \frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+48}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 16000000000000:\\ \;\;\;\;1 + \frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 16000000000000.0)
   (+ 1.0 (/ (- (* -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 <= 16000000000000.0) {
		tmp = 1.0 + (((-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 <= 16000000000000.0d0) then
        tmp = 1.0d0 + ((((-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 <= 16000000000000.0) {
		tmp = 1.0 + (((-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 <= 16000000000000.0:
		tmp = 1.0 + (((-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 <= 16000000000000.0)
		tmp = Float64(1.0 + 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 <= 16000000000000.0)
		tmp = 1.0 + (((-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, 16000000000000.0], N[(1.0 + N[(N[(N[(-0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $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 < 1.6e13

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.6%

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

      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 x around 0 99.5%

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

   if 1.6e13 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow1/2 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 16000000000000:\\ \;\;\;\;1 + \frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}} + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.5e-5)
   (+ (* -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 <= 9.5e-5) {
		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 <= 9.5d-5) 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 <= 9.5e-5) {
		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 <= 9.5e-5:
		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 <= 9.5e-5)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(y / sqrt(x))) + Float64(-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 <= 9.5e-5)
		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, 9.5e-5], N[(N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.1111111111111111 / x), $MachinePrecision]), $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 < 9.5000000000000005e-5

   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* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. div-inv 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0 99.1%

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

   if 9.5000000000000005e-5 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow1/2 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}} + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{if}\;x \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (sqrt (* x 9.0)))))
   (if (<= x 9.5e-5) (- (/ -0.1111111111111111 x) t_0) (- 1.0 t_0))))

C

double code(double x, double y) {
	double t_0 = y / sqrt((x * 9.0));
	double tmp;
	if (x <= 9.5e-5) {
		tmp = (-0.1111111111111111 / x) - t_0;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / sqrt((x * 9.0d0))
    if (x <= 9.5d-5) then
        tmp = ((-0.1111111111111111d0) / x) - t_0
    else
        tmp = 1.0d0 - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / Math.sqrt((x * 9.0));
	double tmp;
	if (x <= 9.5e-5) {
		tmp = (-0.1111111111111111 / x) - t_0;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / math.sqrt((x * 9.0))
	tmp = 0
	if x <= 9.5e-5:
		tmp = (-0.1111111111111111 / x) - t_0
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / sqrt(Float64(x * 9.0)))
	tmp = 0.0
	if (x <= 9.5e-5)
		tmp = Float64(Float64(-0.1111111111111111 / x) - t_0);
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / sqrt((x * 9.0));
	tmp = 0.0;
	if (x <= 9.5e-5)
		tmp = (-0.1111111111111111 / x) - t_0;
	else
		tmp = 1.0 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9.5e-5], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000005e-5

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 99.2%

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

   if 9.5000000000000005e-5 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow1/2 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 11: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $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. Final simplification 99.6%

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

Alternative 12: 63.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1 - {\left(x \cdot 9\right)}^{-1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (pow (* x 9.0) -1.0)))

C

double code(double x, double y) {
	return 1.0 - pow((x * 9.0), -1.0);
}

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 - math.pow((x * 9.0), -1.0)

Julia

function code(x, y)
	return Float64(1.0 - (Float64(x * 9.0) ^ -1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - ((x * 9.0) ^ -1.0);
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.7%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 58.7%

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

   1. div-inv 58.7%

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

   2. metadata-eval 58.7%

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

   3. cancel-sign-sub-inv 58.7%

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

   4. *-commutative 58.7%

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

   5. metadata-eval 58.7%

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

   6. div-inv 58.7%

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

   7. associate-/r* 58.8%

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

   8. add-sqr-sqrt 58.6%

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

   9. sqrt-unprod 44.0%

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

   10. *-commutative 44.0%

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

   11. associate-/r* 44.0%

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

   12. metadata-eval 44.0%

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

   13. *-commutative 44.0%

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

   14. associate-/r* 44.0%

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

   15. metadata-eval 44.0%

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

   16. frac-times 44.0%

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

   17. metadata-eval 44.0%

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

   18. metadata-eval 44.0%

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

   19. frac-times 44.0%

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

   20. sqrt-unprod 0.0%

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

   21. add-sqr-sqrt 26.0%

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

7. Applied egg-rr 26.0%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 44.0%

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

   3. frac-times 44.0%

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

   4. metadata-eval 44.0%

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

   5. metadata-eval 44.0%

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

   6. frac-times 44.0%

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

   7. clear-num 44.0%

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

   8. clear-num 44.0%

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

   9. frac-times 44.0%

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

   10. div-inv 44.0%

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

   11. metadata-eval 44.0%

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

   12. div-inv 44.0%

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

   13. metadata-eval 44.0%

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

   14. frac-times 44.0%

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

   15. sqrt-unprod 58.6%

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

   16. add-sqr-sqrt 58.8%

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

   17. inv-pow 58.8%

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

9. Applied egg-rr 58.8%

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

10. Final simplification 58.8%

    \[\leadsto 1 - {\left(x \cdot 9\right)}^{-1} \]
11. Add Preprocessing

Alternative 13: 63.1% 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. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.7%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 58.7%

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

   1. clear-num 58.7%

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

   2. associate-/r/ 58.7%

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

7. Applied egg-rr 58.7%

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

8. Final simplification 58.7%

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

Alternative 14: 63.1% 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* 99.7%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 58.7%

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

6. Final simplification 58.7%

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

Alternative 15: 31.5% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. distribute-neg-in 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. sub-neg 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.7%

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

   10. fma-neg 99.7%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 58.7%

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

   1. div-inv 58.7%

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

   2. metadata-eval 58.7%

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

   3. cancel-sign-sub-inv 58.7%

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

   4. *-commutative 58.7%

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

   5. metadata-eval 58.7%

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

   6. div-inv 58.7%

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

   7. associate-/r* 58.8%

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

   8. add-sqr-sqrt 58.6%

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

   9. sqrt-unprod 44.0%

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

   10. *-commutative 44.0%

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

   11. associate-/r* 44.0%

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

   12. metadata-eval 44.0%

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

   13. *-commutative 44.0%

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

   14. associate-/r* 44.0%

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

   15. metadata-eval 44.0%

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

   16. frac-times 44.0%

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

   17. metadata-eval 44.0%

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

   18. metadata-eval 44.0%

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

   19. frac-times 44.0%

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

   20. sqrt-unprod 0.0%

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

   21. add-sqr-sqrt 26.0%

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

7. Applied egg-rr 26.0%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 44.0%

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

   3. frac-times 44.0%

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

   4. metadata-eval 44.0%

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

   5. metadata-eval 44.0%

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

   6. frac-times 44.0%

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

   7. clear-num 44.0%

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

   8. clear-num 44.0%

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

   9. frac-times 44.0%

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

   10. div-inv 44.0%

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

   11. metadata-eval 44.0%

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

   12. div-inv 44.0%

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

   13. metadata-eval 44.0%

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

   14. frac-times 44.0%

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

   15. sqrt-unprod 58.6%

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

   16. add-sqr-sqrt 58.8%

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

   17. inv-pow 58.8%

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

9. Applied egg-rr 58.8%

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

10. Taylor expanded in x around inf 26.0%

    \[\leadsto \color{blue}{1} \]

11. Final simplification 26.0%

    \[\leadsto 1 \]
12. 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 2024076 
(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)))))