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

Percentage Accurate: 99.7% → 99.7%

Time: 14.1s

Alternatives: 20

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: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+48}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e+48)
   (+ 1.0 (* y (* -0.3333333333333333 (pow x -0.5))))
   (if (<= y 1.5e+46)
     (+ 1.0 (/ -0.3333333333333333 (* x 3.0)))
     (+ 1.0 (* -0.3333333333333333 (* y (sqrt (/ 1.0 x))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+48) {
		tmp = 1.0 + (y * (-0.3333333333333333 * pow(x, -0.5)));
	} else if (y <= 1.5e+46) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = 1.0 + (-0.3333333333333333 * (y * sqrt((1.0 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9.5d+48)) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) * (x ** (-0.5d0))))
    else if (y <= 1.5d+46) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (x * 3.0d0))
    else
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y * sqrt((1.0d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+48) {
		tmp = 1.0 + (y * (-0.3333333333333333 * Math.pow(x, -0.5)));
	} else if (y <= 1.5e+46) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = 1.0 + (-0.3333333333333333 * (y * Math.sqrt((1.0 / x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e+48:
		tmp = 1.0 + (y * (-0.3333333333333333 * math.pow(x, -0.5)))
	elif y <= 1.5e+46:
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0))
	else:
		tmp = 1.0 + (-0.3333333333333333 * (y * math.sqrt((1.0 / x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e+48)
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5))));
	elseif (y <= 1.5e+46)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(x * 3.0)));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.5e+48)
		tmp = 1.0 + (y * (-0.3333333333333333 * (x ^ -0.5)));
	elseif (y <= 1.5e+46)
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	else
		tmp = 1.0 + (-0.3333333333333333 * (y * sqrt((1.0 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e+48], N[(1.0 + N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+46], N[(1.0 + N[(-0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.4999999999999997e48

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      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* 99.4%

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

      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 96.1%

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

      1. associate-*r* 96.1%

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

   7. Simplified 96.1%

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

      1. expm1-log1p-u 50.5%

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

      2. expm1-udef 14.0%

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

      3. inv-pow 14.0%

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

      4. sqrt-pow1 14.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot y \]

      5. metadata-eval 14.0%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot y \]

   9. Applied egg-rr 14.0%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} - 1\right)} \cdot y \]
   10. Step-by-step derivation

       1. expm1-def 50.5%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\right)} \cdot y \]

       2. expm1-log1p 96.1%

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

   11. Simplified 96.1%

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

   if -9.4999999999999997e48 < y < 1.50000000000000012e46

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.5%

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

      1. cancel-sign-sub-inv 97.5%

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

      2. metadata-eval 97.5%

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

      3. associate-*r/ 97.4%

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

      4. metadata-eval 97.4%

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

      5. +-commutative 97.4%

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

   7. Simplified 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 56.4%

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

      3. frac-times 56.4%

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

      4. metadata-eval 56.4%

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

      5. metadata-eval 56.4%

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

      6. frac-times 56.4%

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

      7. sqrt-unprod 56.4%

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

      8. add-sqr-sqrt 56.4%

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

      9. clear-num 56.4%

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

      10. div-inv 56.4%

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

      11. metadata-eval 56.4%

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

      12. inv-pow 56.4%

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

   9. Applied egg-rr 56.4%

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

       1. inv-pow 56.4%

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

       2. add-sqr-sqrt 56.4%

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

       3. associate-/r* 56.4%

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

       4. metadata-eval 56.4%

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

       5. sqrt-div 56.4%

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

       6. inv-pow 56.4%

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

       7. inv-pow 56.4%

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

       8. metadata-eval 56.4%

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

       9. div-inv 56.4%

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

       10. clear-num 56.4%

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

       11. add-sqr-sqrt 56.4%

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

       12. metadata-eval 56.4%

           \[\leadsto \frac{\sqrt{\frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       13. frac-times 56.4%

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       14. sqrt-prod 0.0%

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       15. add-sqr-sqrt 97.4%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       16. *-commutative 97.4%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{9 \cdot x}}} + 1 \]

       17. sqrt-prod 97.3%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{9} \cdot \sqrt{x}}} + 1 \]

       18. metadata-eval 97.3%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{3} \cdot \sqrt{x}} + 1 \]

   11. Applied egg-rr 97.3%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\sqrt{x}}}{3 \cdot \sqrt{x}}} + 1 \]
   12. Step-by-step derivation

       1. associate-/l/ 97.4%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\left(3 \cdot \sqrt{x}\right) \cdot \sqrt{x}}} + 1 \]

       2. *-commutative 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \left(3 \cdot \sqrt{x}\right)}} + 1 \]

       3. metadata-eval 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(\color{blue}{\left(-3 \cdot -1\right)} \cdot \sqrt{x}\right)} + 1 \]

       4. associate-*r* 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \color{blue}{\left(-3 \cdot \left(-1 \cdot \sqrt{x}\right)\right)}} + 1 \]

       5. neg-mul-1 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(-3 \cdot \color{blue}{\left(-\sqrt{x}\right)}\right)} + 1 \]

       6. associate-*l* 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(-\sqrt{x}\right)}} + 1 \]

       7. distribute-rgt-neg-out 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-\left(\sqrt{x} \cdot -3\right) \cdot \sqrt{x}}} + 1 \]

       8. *-commutative 97.4%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot -3\right)}} + 1 \]

       9. associate-*r* 97.4%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot -3}} + 1 \]

       10. rem-square-sqrt 97.5%

           \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{x} \cdot -3} + 1 \]

       11. distribute-rgt-neg-in 97.5%

           \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot \left(--3\right)}} + 1 \]

       12. metadata-eval 97.5%

           \[\leadsto \frac{-0.3333333333333333}{x \cdot \color{blue}{3}} + 1 \]

   13. Simplified 97.5%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot 3}} + 1 \]

   if 1.50000000000000012e46 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.3%

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

      10. fma-neg 99.3%

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

      11. associate-/r* 99.3%

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

      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 91.1%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+48}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \mathbf{if}\;y \leq -8 \cdot 10^{+48}:\\ \;\;\;\;1 + t_0 \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+44}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (if (<= y -8e+48)
     (+ 1.0 (* t_0 (* y -0.3333333333333333)))
     (if (<= y 6.2e+44)
       (+ 1.0 (/ -0.3333333333333333 (* x 3.0)))
       (+ 1.0 (* -0.3333333333333333 (* y t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((1.0 / x));
	double tmp;
	if (y <= -8e+48) {
		tmp = 1.0 + (t_0 * (y * -0.3333333333333333));
	} else if (y <= 6.2e+44) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = 1.0 + (-0.3333333333333333 * (y * 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 = sqrt((1.0d0 / x))
    if (y <= (-8d+48)) then
        tmp = 1.0d0 + (t_0 * (y * (-0.3333333333333333d0)))
    else if (y <= 6.2d+44) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (x * 3.0d0))
    else
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((1.0 / x));
	double tmp;
	if (y <= -8e+48) {
		tmp = 1.0 + (t_0 * (y * -0.3333333333333333));
	} else if (y <= 6.2e+44) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = 1.0 + (-0.3333333333333333 * (y * t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((1.0 / x))
	tmp = 0
	if y <= -8e+48:
		tmp = 1.0 + (t_0 * (y * -0.3333333333333333))
	elif y <= 6.2e+44:
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0))
	else:
		tmp = 1.0 + (-0.3333333333333333 * (y * t_0))
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(1.0 / x))
	tmp = 0.0
	if (y <= -8e+48)
		tmp = Float64(1.0 + Float64(t_0 * Float64(y * -0.3333333333333333)));
	elseif (y <= 6.2e+44)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(x * 3.0)));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((1.0 / x));
	tmp = 0.0;
	if (y <= -8e+48)
		tmp = 1.0 + (t_0 * (y * -0.3333333333333333));
	elseif (y <= 6.2e+44)
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	else
		tmp = 1.0 + (-0.3333333333333333 * (y * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -8e+48], N[(1.0 + N[(t$95$0 * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+44], N[(1.0 + N[(-0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000035e48

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r/ 99.5%

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

      10. fma-neg 99.5%

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

      11. associate-/r* 99.5%

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

      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* 99.4%

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

      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 96.1%

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

      1. *-commutative 96.1%

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

      2. associate-*l* 96.1%

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

      3. *-commutative 96.1%

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

   7. Simplified 96.1%

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

   if -8.00000000000000035e48 < y < 6.19999999999999991e44

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.5%

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

      1. cancel-sign-sub-inv 97.5%

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

      2. metadata-eval 97.5%

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

      3. associate-*r/ 97.4%

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

      4. metadata-eval 97.4%

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

      5. +-commutative 97.4%

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

   7. Simplified 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 56.4%

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

      3. frac-times 56.4%

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

      4. metadata-eval 56.4%

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

      5. metadata-eval 56.4%

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

      6. frac-times 56.4%

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

      7. sqrt-unprod 56.4%

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

      8. add-sqr-sqrt 56.4%

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

      9. clear-num 56.4%

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

      10. div-inv 56.4%

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

      11. metadata-eval 56.4%

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

      12. inv-pow 56.4%

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

   9. Applied egg-rr 56.4%

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

       1. inv-pow 56.4%

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

       2. add-sqr-sqrt 56.4%

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

       3. associate-/r* 56.4%

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

       4. metadata-eval 56.4%

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

       5. sqrt-div 56.4%

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

       6. inv-pow 56.4%

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

       7. inv-pow 56.4%

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

       8. metadata-eval 56.4%

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

       9. div-inv 56.4%

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

       10. clear-num 56.4%

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

       11. add-sqr-sqrt 56.4%

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

       12. metadata-eval 56.4%

           \[\leadsto \frac{\sqrt{\frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       13. frac-times 56.4%

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       14. sqrt-prod 0.0%

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       15. add-sqr-sqrt 97.4%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       16. *-commutative 97.4%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{9 \cdot x}}} + 1 \]

       17. sqrt-prod 97.3%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{9} \cdot \sqrt{x}}} + 1 \]

       18. metadata-eval 97.3%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{3} \cdot \sqrt{x}} + 1 \]

   11. Applied egg-rr 97.3%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\sqrt{x}}}{3 \cdot \sqrt{x}}} + 1 \]
   12. Step-by-step derivation

       1. associate-/l/ 97.4%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\left(3 \cdot \sqrt{x}\right) \cdot \sqrt{x}}} + 1 \]

       2. *-commutative 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \left(3 \cdot \sqrt{x}\right)}} + 1 \]

       3. metadata-eval 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(\color{blue}{\left(-3 \cdot -1\right)} \cdot \sqrt{x}\right)} + 1 \]

       4. associate-*r* 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \color{blue}{\left(-3 \cdot \left(-1 \cdot \sqrt{x}\right)\right)}} + 1 \]

       5. neg-mul-1 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(-3 \cdot \color{blue}{\left(-\sqrt{x}\right)}\right)} + 1 \]

       6. associate-*l* 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(-\sqrt{x}\right)}} + 1 \]

       7. distribute-rgt-neg-out 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-\left(\sqrt{x} \cdot -3\right) \cdot \sqrt{x}}} + 1 \]

       8. *-commutative 97.4%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot -3\right)}} + 1 \]

       9. associate-*r* 97.4%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot -3}} + 1 \]

       10. rem-square-sqrt 97.5%

           \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{x} \cdot -3} + 1 \]

       11. distribute-rgt-neg-in 97.5%

           \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot \left(--3\right)}} + 1 \]

       12. metadata-eval 97.5%

           \[\leadsto \frac{-0.3333333333333333}{x \cdot \color{blue}{3}} + 1 \]

   13. Simplified 97.5%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot 3}} + 1 \]

   if 6.19999999999999991e44 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.3%

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

      10. fma-neg 99.3%

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

      11. associate-/r* 99.3%

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

      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 91.1%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+48}:\\ \;\;\;\;1 + \sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+44}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+48} \lor \neg \left(y \leq 3.4 \cdot 10^{+45}\right):\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.2e+48) (not (<= y 3.4e+45)))
   (+ 1.0 (* y (* -0.3333333333333333 (pow x -0.5))))
   (+ 1.0 (/ -0.3333333333333333 (* x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+48) || !(y <= 3.4e+45)) {
		tmp = 1.0 + (y * (-0.3333333333333333 * pow(x, -0.5)));
	} else {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.2d+48)) .or. (.not. (y <= 3.4d+45))) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) * (x ** (-0.5d0))))
    else
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (x * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.2e+48) || !(y <= 3.4e+45)) {
		tmp = 1.0 + (y * (-0.3333333333333333 * Math.pow(x, -0.5)));
	} else {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.2e+48) or not (y <= 3.4e+45):
		tmp = 1.0 + (y * (-0.3333333333333333 * math.pow(x, -0.5)))
	else:
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.2e+48) || !(y <= 3.4e+45))
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5))));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(x * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.2e+48) || ~((y <= 3.4e+45)))
		tmp = 1.0 + (y * (-0.3333333333333333 * (x ^ -0.5)));
	else
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.2e+48], N[Not[LessEqual[y, 3.4e+45]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1999999999999997e48 or 3.4e45 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r- 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.4%

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

      10. fma-neg 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 93.8%

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

      1. associate-*r* 93.7%

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

   7. Simplified 93.7%

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

      1. expm1-log1p-u 51.9%

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

      2. expm1-udef 11.6%

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

      3. inv-pow 11.6%

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

      4. sqrt-pow1 11.6%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot y \]

      5. metadata-eval 11.6%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot y \]

   9. Applied egg-rr 11.6%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} - 1\right)} \cdot y \]
   10. Step-by-step derivation

       1. expm1-def 51.9%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\right)} \cdot y \]

       2. expm1-log1p 93.7%

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

   11. Simplified 93.7%

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

   if -4.1999999999999997e48 < y < 3.4e45

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.5%

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

      1. cancel-sign-sub-inv 97.5%

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

      2. metadata-eval 97.5%

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

      3. associate-*r/ 97.4%

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

      4. metadata-eval 97.4%

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

      5. +-commutative 97.4%

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

   7. Simplified 97.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 56.4%

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

      3. frac-times 56.4%

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

      4. metadata-eval 56.4%

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

      5. metadata-eval 56.4%

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

      6. frac-times 56.4%

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

      7. sqrt-unprod 56.4%

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

      8. add-sqr-sqrt 56.4%

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

      9. clear-num 56.4%

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

      10. div-inv 56.4%

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

      11. metadata-eval 56.4%

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

      12. inv-pow 56.4%

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

   9. Applied egg-rr 56.4%

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

       1. inv-pow 56.4%

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

       2. add-sqr-sqrt 56.4%

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

       3. associate-/r* 56.4%

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

       4. metadata-eval 56.4%

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

       5. sqrt-div 56.4%

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

       6. inv-pow 56.4%

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

       7. inv-pow 56.4%

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

       8. metadata-eval 56.4%

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

       9. div-inv 56.4%

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

       10. clear-num 56.4%

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

       11. add-sqr-sqrt 56.4%

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

       12. metadata-eval 56.4%

           \[\leadsto \frac{\sqrt{\frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       13. frac-times 56.4%

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       14. sqrt-prod 0.0%

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       15. add-sqr-sqrt 97.4%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       16. *-commutative 97.4%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{9 \cdot x}}} + 1 \]

       17. sqrt-prod 97.3%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{9} \cdot \sqrt{x}}} + 1 \]

       18. metadata-eval 97.3%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{3} \cdot \sqrt{x}} + 1 \]

   11. Applied egg-rr 97.3%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\sqrt{x}}}{3 \cdot \sqrt{x}}} + 1 \]
   12. Step-by-step derivation

       1. associate-/l/ 97.4%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\left(3 \cdot \sqrt{x}\right) \cdot \sqrt{x}}} + 1 \]

       2. *-commutative 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \left(3 \cdot \sqrt{x}\right)}} + 1 \]

       3. metadata-eval 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(\color{blue}{\left(-3 \cdot -1\right)} \cdot \sqrt{x}\right)} + 1 \]

       4. associate-*r* 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \color{blue}{\left(-3 \cdot \left(-1 \cdot \sqrt{x}\right)\right)}} + 1 \]

       5. neg-mul-1 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(-3 \cdot \color{blue}{\left(-\sqrt{x}\right)}\right)} + 1 \]

       6. associate-*l* 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(-\sqrt{x}\right)}} + 1 \]

       7. distribute-rgt-neg-out 97.4%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-\left(\sqrt{x} \cdot -3\right) \cdot \sqrt{x}}} + 1 \]

       8. *-commutative 97.4%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot -3\right)}} + 1 \]

       9. associate-*r* 97.4%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot -3}} + 1 \]

       10. rem-square-sqrt 97.5%

           \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{x} \cdot -3} + 1 \]

       11. distribute-rgt-neg-in 97.5%

           \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot \left(--3\right)}} + 1 \]

       12. metadata-eval 97.5%

           \[\leadsto \frac{-0.3333333333333333}{x \cdot \color{blue}{3}} + 1 \]

   13. Simplified 97.5%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot 3}} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+48} \lor \neg \left(y \leq 3.4 \cdot 10^{+45}\right):\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+66}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.6e+48)
   (/ -0.3333333333333333 (/ (sqrt x) y))
   (if (<= y 1.52e+66)
     (+ 1.0 (/ -0.3333333333333333 (* x 3.0)))
     (* -0.3333333333333333 (* y (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.6e+48) {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	} else if (y <= 1.52e+66) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.6d+48)) then
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    else if (y <= 1.52d+66) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (x * 3.0d0))
    else
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.6e+48) {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	} else if (y <= 1.52e+66) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.6e+48:
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	elif y <= 1.52e+66:
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0))
	else:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.6e+48)
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	elseif (y <= 1.52e+66)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(x * 3.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.6e+48)
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	elseif (y <= 1.52e+66)
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	else
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.6e+48], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.52e+66], N[(1.0 + N[(-0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.60000000000000001e48

   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 y around inf 87.3%

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

      1. *-commutative 87.3%

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

   9. Simplified 87.3%

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

       1. *-commutative 87.3%

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

       2. sqrt-div 87.3%

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

       3. metadata-eval 87.3%

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

       4. div-inv 87.4%

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

       5. *-commutative 87.4%

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

       6. clear-num 87.3%

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

       7. un-div-inv 87.4%

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

   11. Applied egg-rr 87.4%

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

   if -7.60000000000000001e48 < y < 1.52000000000000004e66

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 95.1%

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

      1. cancel-sign-sub-inv 95.1%

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

      2. metadata-eval 95.1%

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

      3. associate-*r/ 95.1%

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

      4. metadata-eval 95.1%

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

      5. +-commutative 95.1%

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

   7. Simplified 95.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.4%

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

      3. frac-times 54.4%

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

      4. metadata-eval 54.4%

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

      5. metadata-eval 54.4%

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

      6. frac-times 54.4%

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

      7. sqrt-unprod 54.5%

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

      8. add-sqr-sqrt 54.5%

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

      9. clear-num 54.5%

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

      10. div-inv 54.5%

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

      11. metadata-eval 54.5%

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

      12. inv-pow 54.5%

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

   9. Applied egg-rr 54.5%

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

       1. inv-pow 54.5%

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

       2. add-sqr-sqrt 54.5%

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

       3. associate-/r* 54.5%

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

       4. metadata-eval 54.5%

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

       5. sqrt-div 54.5%

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

       6. inv-pow 54.5%

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

       7. inv-pow 54.5%

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

       8. metadata-eval 54.5%

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

       9. div-inv 54.5%

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

       10. clear-num 54.5%

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

       11. add-sqr-sqrt 54.5%

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

       12. metadata-eval 54.5%

           \[\leadsto \frac{\sqrt{\frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       13. frac-times 54.5%

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       14. sqrt-prod 0.0%

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       15. add-sqr-sqrt 95.1%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       16. *-commutative 95.1%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{9 \cdot x}}} + 1 \]

       17. sqrt-prod 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{9} \cdot \sqrt{x}}} + 1 \]

       18. metadata-eval 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{3} \cdot \sqrt{x}} + 1 \]

   11. Applied egg-rr 95.0%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\sqrt{x}}}{3 \cdot \sqrt{x}}} + 1 \]
   12. Step-by-step derivation

       1. associate-/l/ 95.0%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\left(3 \cdot \sqrt{x}\right) \cdot \sqrt{x}}} + 1 \]

       2. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \left(3 \cdot \sqrt{x}\right)}} + 1 \]

       3. metadata-eval 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(\color{blue}{\left(-3 \cdot -1\right)} \cdot \sqrt{x}\right)} + 1 \]

       4. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \color{blue}{\left(-3 \cdot \left(-1 \cdot \sqrt{x}\right)\right)}} + 1 \]

       5. neg-mul-1 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(-3 \cdot \color{blue}{\left(-\sqrt{x}\right)}\right)} + 1 \]

       6. associate-*l* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(-\sqrt{x}\right)}} + 1 \]

       7. distribute-rgt-neg-out 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-\left(\sqrt{x} \cdot -3\right) \cdot \sqrt{x}}} + 1 \]

       8. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot -3\right)}} + 1 \]

       9. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot -3}} + 1 \]

       10. rem-square-sqrt 95.1%

           \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{x} \cdot -3} + 1 \]

       11. distribute-rgt-neg-in 95.1%

           \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot \left(--3\right)}} + 1 \]

       12. metadata-eval 95.1%

           \[\leadsto \frac{-0.3333333333333333}{x \cdot \color{blue}{3}} + 1 \]

   13. Simplified 95.1%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot 3}} + 1 \]

   if 1.52000000000000004e66 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around inf 88.8%

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

      1. *-commutative 88.8%

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

   9. Simplified 88.8%

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

       1. inv-pow 88.8%

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

       2. sqrt-pow1 88.9%

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

       3. metadata-eval 88.9%

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

       4. expm1-log1p-u 87.9%

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \cdot -0.3333333333333333 \]

       5. expm1-udef 41.7%

          \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \cdot -0.3333333333333333 \]

   11. Applied egg-rr 41.7%

       \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \cdot -0.3333333333333333 \]
   12. Step-by-step derivation

       1. expm1-def 87.9%

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \cdot -0.3333333333333333 \]

       2. expm1-log1p 88.9%

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

   13. Simplified 88.9%

       \[\leadsto \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \cdot -0.3333333333333333 \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+66}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+66}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+49)
   (* y (* -0.3333333333333333 (sqrt (/ 1.0 x))))
   (if (<= y 2.1e+66)
     (+ 1.0 (/ -0.3333333333333333 (* x 3.0)))
     (* -0.3333333333333333 (* y (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+49) {
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	} else if (y <= 2.1e+66) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.45d+49)) then
        tmp = y * ((-0.3333333333333333d0) * sqrt((1.0d0 / x)))
    else if (y <= 2.1d+66) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (x * 3.0d0))
    else
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+49) {
		tmp = y * (-0.3333333333333333 * Math.sqrt((1.0 / x)));
	} else if (y <= 2.1e+66) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+49:
		tmp = y * (-0.3333333333333333 * math.sqrt((1.0 / x)))
	elif y <= 2.1e+66:
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0))
	else:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+49)
		tmp = Float64(y * Float64(-0.3333333333333333 * sqrt(Float64(1.0 / x))));
	elseif (y <= 2.1e+66)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(x * 3.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+49)
		tmp = y * (-0.3333333333333333 * sqrt((1.0 / x)));
	elseif (y <= 2.1e+66)
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	else
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+49], N[(y * N[(-0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+66], N[(1.0 + N[(-0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45e49

   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 y around inf 87.3%

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

      1. *-commutative 87.3%

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

   9. Simplified 87.3%

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

   10. Taylor expanded in y around 0 87.3%

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

       1. *-commutative 87.3%

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

       2. *-commutative 87.3%

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

       3. associate-*l* 87.4%

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

   12. Simplified 87.4%

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

   if -1.45e49 < y < 2.10000000000000005e66

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 95.1%

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

      1. cancel-sign-sub-inv 95.1%

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

      2. metadata-eval 95.1%

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

      3. associate-*r/ 95.1%

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

      4. metadata-eval 95.1%

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

      5. +-commutative 95.1%

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

   7. Simplified 95.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.4%

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

      3. frac-times 54.4%

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

      4. metadata-eval 54.4%

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

      5. metadata-eval 54.4%

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

      6. frac-times 54.4%

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

      7. sqrt-unprod 54.5%

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

      8. add-sqr-sqrt 54.5%

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

      9. clear-num 54.5%

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

      10. div-inv 54.5%

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

      11. metadata-eval 54.5%

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

      12. inv-pow 54.5%

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

   9. Applied egg-rr 54.5%

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

       1. inv-pow 54.5%

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

       2. add-sqr-sqrt 54.5%

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

       3. associate-/r* 54.5%

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

       4. metadata-eval 54.5%

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

       5. sqrt-div 54.5%

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

       6. inv-pow 54.5%

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

       7. inv-pow 54.5%

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

       8. metadata-eval 54.5%

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

       9. div-inv 54.5%

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

       10. clear-num 54.5%

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

       11. add-sqr-sqrt 54.5%

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

       12. metadata-eval 54.5%

           \[\leadsto \frac{\sqrt{\frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       13. frac-times 54.5%

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       14. sqrt-prod 0.0%

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       15. add-sqr-sqrt 95.1%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       16. *-commutative 95.1%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{9 \cdot x}}} + 1 \]

       17. sqrt-prod 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{9} \cdot \sqrt{x}}} + 1 \]

       18. metadata-eval 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{3} \cdot \sqrt{x}} + 1 \]

   11. Applied egg-rr 95.0%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\sqrt{x}}}{3 \cdot \sqrt{x}}} + 1 \]
   12. Step-by-step derivation

       1. associate-/l/ 95.0%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\left(3 \cdot \sqrt{x}\right) \cdot \sqrt{x}}} + 1 \]

       2. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \left(3 \cdot \sqrt{x}\right)}} + 1 \]

       3. metadata-eval 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(\color{blue}{\left(-3 \cdot -1\right)} \cdot \sqrt{x}\right)} + 1 \]

       4. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \color{blue}{\left(-3 \cdot \left(-1 \cdot \sqrt{x}\right)\right)}} + 1 \]

       5. neg-mul-1 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(-3 \cdot \color{blue}{\left(-\sqrt{x}\right)}\right)} + 1 \]

       6. associate-*l* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(-\sqrt{x}\right)}} + 1 \]

       7. distribute-rgt-neg-out 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-\left(\sqrt{x} \cdot -3\right) \cdot \sqrt{x}}} + 1 \]

       8. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot -3\right)}} + 1 \]

       9. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot -3}} + 1 \]

       10. rem-square-sqrt 95.1%

           \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{x} \cdot -3} + 1 \]

       11. distribute-rgt-neg-in 95.1%

           \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot \left(--3\right)}} + 1 \]

       12. metadata-eval 95.1%

           \[\leadsto \frac{-0.3333333333333333}{x \cdot \color{blue}{3}} + 1 \]

   13. Simplified 95.1%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot 3}} + 1 \]

   if 2.10000000000000005e66 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around inf 88.8%

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

      1. *-commutative 88.8%

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

   9. Simplified 88.8%

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

       1. inv-pow 88.8%

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

       2. sqrt-pow1 88.9%

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

       3. metadata-eval 88.9%

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

       4. expm1-log1p-u 87.9%

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \cdot -0.3333333333333333 \]

       5. expm1-udef 41.7%

          \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \cdot -0.3333333333333333 \]

   11. Applied egg-rr 41.7%

       \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \cdot -0.3333333333333333 \]
   12. Step-by-step derivation

       1. expm1-def 87.9%

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \cdot -0.3333333333333333 \]

       2. expm1-log1p 88.9%

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

   13. Simplified 88.9%

       \[\leadsto \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \cdot -0.3333333333333333 \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+66}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+69}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+49)
   (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))
   (if (<= y 4.6e+69)
     (+ 1.0 (/ -0.3333333333333333 (* x 3.0)))
     (* -0.3333333333333333 (* y (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+49) {
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} else if (y <= 4.6e+69) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = -0.3333333333333333 * (y * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.45d+49)) then
        tmp = sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0))
    else if (y <= 4.6d+69) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (x * 3.0d0))
    else
        tmp = (-0.3333333333333333d0) * (y * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+49) {
		tmp = Math.sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} else if (y <= 4.6e+69) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = -0.3333333333333333 * (y * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+49:
		tmp = math.sqrt((1.0 / x)) * (y * -0.3333333333333333)
	elif y <= 4.6e+69:
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0))
	else:
		tmp = -0.3333333333333333 * (y * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+49)
		tmp = Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333));
	elseif (y <= 4.6e+69)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(x * 3.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+49)
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	elseif (y <= 4.6e+69)
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	else
		tmp = -0.3333333333333333 * (y * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+49], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+69], N[(1.0 + N[(-0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45e49

   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 y around inf 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-*l* 87.4%

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

   9. Simplified 87.4%

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

   if -1.45e49 < y < 4.60000000000000033e69

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 95.1%

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

      1. cancel-sign-sub-inv 95.1%

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

      2. metadata-eval 95.1%

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

      3. associate-*r/ 95.1%

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

      4. metadata-eval 95.1%

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

      5. +-commutative 95.1%

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

   7. Simplified 95.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.4%

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

      3. frac-times 54.4%

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

      4. metadata-eval 54.4%

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

      5. metadata-eval 54.4%

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

      6. frac-times 54.4%

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

      7. sqrt-unprod 54.5%

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

      8. add-sqr-sqrt 54.5%

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

      9. clear-num 54.5%

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

      10. div-inv 54.5%

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

      11. metadata-eval 54.5%

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

      12. inv-pow 54.5%

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

   9. Applied egg-rr 54.5%

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

       1. inv-pow 54.5%

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

       2. add-sqr-sqrt 54.5%

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

       3. associate-/r* 54.5%

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

       4. metadata-eval 54.5%

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

       5. sqrt-div 54.5%

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

       6. inv-pow 54.5%

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

       7. inv-pow 54.5%

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

       8. metadata-eval 54.5%

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

       9. div-inv 54.5%

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

       10. clear-num 54.5%

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

       11. add-sqr-sqrt 54.5%

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

       12. metadata-eval 54.5%

           \[\leadsto \frac{\sqrt{\frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       13. frac-times 54.5%

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       14. sqrt-prod 0.0%

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       15. add-sqr-sqrt 95.1%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       16. *-commutative 95.1%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{9 \cdot x}}} + 1 \]

       17. sqrt-prod 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{9} \cdot \sqrt{x}}} + 1 \]

       18. metadata-eval 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{3} \cdot \sqrt{x}} + 1 \]

   11. Applied egg-rr 95.0%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\sqrt{x}}}{3 \cdot \sqrt{x}}} + 1 \]
   12. Step-by-step derivation

       1. associate-/l/ 95.0%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\left(3 \cdot \sqrt{x}\right) \cdot \sqrt{x}}} + 1 \]

       2. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \left(3 \cdot \sqrt{x}\right)}} + 1 \]

       3. metadata-eval 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(\color{blue}{\left(-3 \cdot -1\right)} \cdot \sqrt{x}\right)} + 1 \]

       4. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \color{blue}{\left(-3 \cdot \left(-1 \cdot \sqrt{x}\right)\right)}} + 1 \]

       5. neg-mul-1 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(-3 \cdot \color{blue}{\left(-\sqrt{x}\right)}\right)} + 1 \]

       6. associate-*l* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(-\sqrt{x}\right)}} + 1 \]

       7. distribute-rgt-neg-out 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-\left(\sqrt{x} \cdot -3\right) \cdot \sqrt{x}}} + 1 \]

       8. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot -3\right)}} + 1 \]

       9. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot -3}} + 1 \]

       10. rem-square-sqrt 95.1%

           \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{x} \cdot -3} + 1 \]

       11. distribute-rgt-neg-in 95.1%

           \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot \left(--3\right)}} + 1 \]

       12. metadata-eval 95.1%

           \[\leadsto \frac{-0.3333333333333333}{x \cdot \color{blue}{3}} + 1 \]

   13. Simplified 95.1%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot 3}} + 1 \]

   if 4.60000000000000033e69 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around inf 88.8%

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

      1. *-commutative 88.8%

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

   9. Simplified 88.8%

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

       1. inv-pow 88.8%

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

       2. sqrt-pow1 88.9%

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

       3. metadata-eval 88.9%

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

       4. expm1-log1p-u 87.9%

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \cdot -0.3333333333333333 \]

       5. expm1-udef 41.7%

          \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \cdot -0.3333333333333333 \]

   11. Applied egg-rr 41.7%

       \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \cdot y\right) \cdot -0.3333333333333333 \]
   12. Step-by-step derivation

       1. expm1-def 87.9%

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \cdot y\right) \cdot -0.3333333333333333 \]

       2. expm1-log1p 88.9%

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

   13. Simplified 88.9%

       \[\leadsto \left(\color{blue}{{x}^{-0.5}} \cdot y\right) \cdot -0.3333333333333333 \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+69}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49} \lor \neg \left(y \leq 1.3 \cdot 10^{+66}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.45e+49) (not (<= y 1.3e+66)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ -0.3333333333333333 (* x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e+49) || !(y <= 1.3e+66)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.45d+49)) .or. (.not. (y <= 1.3d+66))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (x * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e+49) || !(y <= 1.3e+66)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.45e+49) or not (y <= 1.3e+66):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.45e+49) || !(y <= 1.3e+66))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(x * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.45e+49) || ~((y <= 1.3e+66)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.45e+49], N[Not[LessEqual[y, 1.3e+66]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e49 or 1.30000000000000006e66 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around inf 87.9%

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

      1. *-commutative 87.9%

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

   9. Simplified 87.9%

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

       1. *-commutative 87.9%

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

       2. sqrt-div 87.9%

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

       3. metadata-eval 87.9%

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

       4. div-inv 87.9%

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

       5. expm1-log1p-u 34.5%

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

       6. expm1-udef 34.4%

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

   11. Applied egg-rr 34.4%

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

       1. expm1-def 34.5%

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

       2. expm1-log1p 87.9%

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

   13. Simplified 87.9%

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

   if -1.45e49 < y < 1.30000000000000006e66

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 95.1%

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

      1. cancel-sign-sub-inv 95.1%

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

      2. metadata-eval 95.1%

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

      3. associate-*r/ 95.1%

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

      4. metadata-eval 95.1%

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

      5. +-commutative 95.1%

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

   7. Simplified 95.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.4%

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

      3. frac-times 54.4%

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

      4. metadata-eval 54.4%

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

      5. metadata-eval 54.4%

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

      6. frac-times 54.4%

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

      7. sqrt-unprod 54.5%

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

      8. add-sqr-sqrt 54.5%

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

      9. clear-num 54.5%

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

      10. div-inv 54.5%

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

      11. metadata-eval 54.5%

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

      12. inv-pow 54.5%

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

   9. Applied egg-rr 54.5%

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

       1. inv-pow 54.5%

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

       2. add-sqr-sqrt 54.5%

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

       3. associate-/r* 54.5%

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

       4. metadata-eval 54.5%

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

       5. sqrt-div 54.5%

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

       6. inv-pow 54.5%

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

       7. inv-pow 54.5%

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

       8. metadata-eval 54.5%

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

       9. div-inv 54.5%

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

       10. clear-num 54.5%

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

       11. add-sqr-sqrt 54.5%

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

       12. metadata-eval 54.5%

           \[\leadsto \frac{\sqrt{\frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       13. frac-times 54.5%

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       14. sqrt-prod 0.0%

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       15. add-sqr-sqrt 95.1%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       16. *-commutative 95.1%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{9 \cdot x}}} + 1 \]

       17. sqrt-prod 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{9} \cdot \sqrt{x}}} + 1 \]

       18. metadata-eval 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{3} \cdot \sqrt{x}} + 1 \]

   11. Applied egg-rr 95.0%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\sqrt{x}}}{3 \cdot \sqrt{x}}} + 1 \]
   12. Step-by-step derivation

       1. associate-/l/ 95.0%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\left(3 \cdot \sqrt{x}\right) \cdot \sqrt{x}}} + 1 \]

       2. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \left(3 \cdot \sqrt{x}\right)}} + 1 \]

       3. metadata-eval 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(\color{blue}{\left(-3 \cdot -1\right)} \cdot \sqrt{x}\right)} + 1 \]

       4. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \color{blue}{\left(-3 \cdot \left(-1 \cdot \sqrt{x}\right)\right)}} + 1 \]

       5. neg-mul-1 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(-3 \cdot \color{blue}{\left(-\sqrt{x}\right)}\right)} + 1 \]

       6. associate-*l* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(-\sqrt{x}\right)}} + 1 \]

       7. distribute-rgt-neg-out 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-\left(\sqrt{x} \cdot -3\right) \cdot \sqrt{x}}} + 1 \]

       8. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot -3\right)}} + 1 \]

       9. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot -3}} + 1 \]

       10. rem-square-sqrt 95.1%

           \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{x} \cdot -3} + 1 \]

       11. distribute-rgt-neg-in 95.1%

           \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot \left(--3\right)}} + 1 \]

       12. metadata-eval 95.1%

           \[\leadsto \frac{-0.3333333333333333}{x \cdot \color{blue}{3}} + 1 \]

   13. Simplified 95.1%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot 3}} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49} \lor \neg \left(y \leq 1.3 \cdot 10^{+66}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+69}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+49)
   (/ -0.3333333333333333 (/ (sqrt x) y))
   (if (<= y 9.2e+69)
     (+ 1.0 (/ -0.3333333333333333 (* x 3.0)))
     (* -0.3333333333333333 (/ y (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+49) {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	} else if (y <= 9.2e+69) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = -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 <= (-1.45d+49)) then
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    else if (y <= 9.2d+69) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (x * 3.0d0))
    else
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+49) {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	} else if (y <= 9.2e+69) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+49:
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	elif y <= 9.2e+69:
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0))
	else:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+49)
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	elseif (y <= 9.2e+69)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(x * 3.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+49)
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	elseif (y <= 9.2e+69)
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	else
		tmp = -0.3333333333333333 * (y / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+49], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e+69], N[(1.0 + N[(-0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45e49

   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 y around inf 87.3%

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

      1. *-commutative 87.3%

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

   9. Simplified 87.3%

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

       1. *-commutative 87.3%

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

       2. sqrt-div 87.3%

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

       3. metadata-eval 87.3%

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

       4. div-inv 87.4%

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

       5. *-commutative 87.4%

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

       6. clear-num 87.3%

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

       7. un-div-inv 87.4%

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

   11. Applied egg-rr 87.4%

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

   if -1.45e49 < y < 9.20000000000000067e69

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 95.1%

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

      1. cancel-sign-sub-inv 95.1%

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

      2. metadata-eval 95.1%

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

      3. associate-*r/ 95.1%

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

      4. metadata-eval 95.1%

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

      5. +-commutative 95.1%

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

   7. Simplified 95.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.4%

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

      3. frac-times 54.4%

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

      4. metadata-eval 54.4%

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

      5. metadata-eval 54.4%

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

      6. frac-times 54.4%

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

      7. sqrt-unprod 54.5%

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

      8. add-sqr-sqrt 54.5%

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

      9. clear-num 54.5%

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

      10. div-inv 54.5%

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

      11. metadata-eval 54.5%

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

      12. inv-pow 54.5%

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

   9. Applied egg-rr 54.5%

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

       1. inv-pow 54.5%

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

       2. add-sqr-sqrt 54.5%

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

       3. associate-/r* 54.5%

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

       4. metadata-eval 54.5%

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

       5. sqrt-div 54.5%

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

       6. inv-pow 54.5%

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

       7. inv-pow 54.5%

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

       8. metadata-eval 54.5%

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

       9. div-inv 54.5%

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

       10. clear-num 54.5%

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

       11. add-sqr-sqrt 54.5%

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

       12. metadata-eval 54.5%

           \[\leadsto \frac{\sqrt{\frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       13. frac-times 54.5%

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       14. sqrt-prod 0.0%

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       15. add-sqr-sqrt 95.1%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       16. *-commutative 95.1%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{9 \cdot x}}} + 1 \]

       17. sqrt-prod 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{9} \cdot \sqrt{x}}} + 1 \]

       18. metadata-eval 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{3} \cdot \sqrt{x}} + 1 \]

   11. Applied egg-rr 95.0%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\sqrt{x}}}{3 \cdot \sqrt{x}}} + 1 \]
   12. Step-by-step derivation

       1. associate-/l/ 95.0%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\left(3 \cdot \sqrt{x}\right) \cdot \sqrt{x}}} + 1 \]

       2. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \left(3 \cdot \sqrt{x}\right)}} + 1 \]

       3. metadata-eval 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(\color{blue}{\left(-3 \cdot -1\right)} \cdot \sqrt{x}\right)} + 1 \]

       4. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \color{blue}{\left(-3 \cdot \left(-1 \cdot \sqrt{x}\right)\right)}} + 1 \]

       5. neg-mul-1 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(-3 \cdot \color{blue}{\left(-\sqrt{x}\right)}\right)} + 1 \]

       6. associate-*l* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(-\sqrt{x}\right)}} + 1 \]

       7. distribute-rgt-neg-out 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-\left(\sqrt{x} \cdot -3\right) \cdot \sqrt{x}}} + 1 \]

       8. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot -3\right)}} + 1 \]

       9. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot -3}} + 1 \]

       10. rem-square-sqrt 95.1%

           \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{x} \cdot -3} + 1 \]

       11. distribute-rgt-neg-in 95.1%

           \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot \left(--3\right)}} + 1 \]

       12. metadata-eval 95.1%

           \[\leadsto \frac{-0.3333333333333333}{x \cdot \color{blue}{3}} + 1 \]

   13. Simplified 95.1%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot 3}} + 1 \]

   if 9.20000000000000067e69 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around inf 88.8%

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

      1. *-commutative 88.8%

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

   9. Simplified 88.8%

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

       1. *-commutative 88.8%

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

       2. sqrt-div 88.8%

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

       3. metadata-eval 88.8%

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

       4. div-inv 88.7%

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

       5. expm1-log1p-u 82.5%

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

       6. expm1-udef 82.5%

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

   11. Applied egg-rr 82.5%

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

       1. expm1-def 82.5%

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

       2. expm1-log1p 88.7%

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

   13. Simplified 88.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+69}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 8.9 \cdot 10^{+67}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+49)
   (/ -0.3333333333333333 (/ (sqrt x) y))
   (if (<= y 8.9e+67)
     (+ 1.0 (/ -0.3333333333333333 (* x 3.0)))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+49) {
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	} else if (y <= 8.9e+67) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.45d+49)) then
        tmp = (-0.3333333333333333d0) / (sqrt(x) / y)
    else if (y <= 8.9d+67) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (x * 3.0d0))
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+49) {
		tmp = -0.3333333333333333 / (Math.sqrt(x) / y);
	} else if (y <= 8.9e+67) {
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+49:
		tmp = -0.3333333333333333 / (math.sqrt(x) / y)
	elif y <= 8.9e+67:
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0))
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+49)
		tmp = Float64(-0.3333333333333333 / Float64(sqrt(x) / y));
	elseif (y <= 8.9e+67)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(x * 3.0)));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+49)
		tmp = -0.3333333333333333 / (sqrt(x) / y);
	elseif (y <= 8.9e+67)
		tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+49], N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.9e+67], N[(1.0 + N[(-0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45e49

   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 y around inf 87.3%

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

      1. *-commutative 87.3%

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

   9. Simplified 87.3%

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

       1. *-commutative 87.3%

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

       2. sqrt-div 87.3%

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

       3. metadata-eval 87.3%

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

       4. div-inv 87.4%

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

       5. *-commutative 87.4%

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

       6. clear-num 87.3%

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

       7. un-div-inv 87.4%

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

   11. Applied egg-rr 87.4%

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

   if -1.45e49 < y < 8.89999999999999983e67

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. fma-neg 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 95.1%

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

      1. cancel-sign-sub-inv 95.1%

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

      2. metadata-eval 95.1%

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

      3. associate-*r/ 95.1%

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

      4. metadata-eval 95.1%

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

      5. +-commutative 95.1%

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

   7. Simplified 95.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 54.4%

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

      3. frac-times 54.4%

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

      4. metadata-eval 54.4%

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

      5. metadata-eval 54.4%

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

      6. frac-times 54.4%

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

      7. sqrt-unprod 54.5%

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

      8. add-sqr-sqrt 54.5%

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

      9. clear-num 54.5%

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

      10. div-inv 54.5%

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

      11. metadata-eval 54.5%

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

      12. inv-pow 54.5%

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

   9. Applied egg-rr 54.5%

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

       1. inv-pow 54.5%

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

       2. add-sqr-sqrt 54.5%

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

       3. associate-/r* 54.5%

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

       4. metadata-eval 54.5%

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

       5. sqrt-div 54.5%

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

       6. inv-pow 54.5%

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

       7. inv-pow 54.5%

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

       8. metadata-eval 54.5%

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

       9. div-inv 54.5%

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

       10. clear-num 54.5%

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

       11. add-sqr-sqrt 54.5%

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

       12. metadata-eval 54.5%

           \[\leadsto \frac{\sqrt{\frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       13. frac-times 54.5%

           \[\leadsto \frac{\sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       14. sqrt-prod 0.0%

           \[\leadsto \frac{\color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

       15. add-sqr-sqrt 95.1%

           \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

       16. *-commutative 95.1%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{9 \cdot x}}} + 1 \]

       17. sqrt-prod 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{9} \cdot \sqrt{x}}} + 1 \]

       18. metadata-eval 95.0%

           \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{3} \cdot \sqrt{x}} + 1 \]

   11. Applied egg-rr 95.0%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\sqrt{x}}}{3 \cdot \sqrt{x}}} + 1 \]
   12. Step-by-step derivation

       1. associate-/l/ 95.0%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\left(3 \cdot \sqrt{x}\right) \cdot \sqrt{x}}} + 1 \]

       2. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \left(3 \cdot \sqrt{x}\right)}} + 1 \]

       3. metadata-eval 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(\color{blue}{\left(-3 \cdot -1\right)} \cdot \sqrt{x}\right)} + 1 \]

       4. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \color{blue}{\left(-3 \cdot \left(-1 \cdot \sqrt{x}\right)\right)}} + 1 \]

       5. neg-mul-1 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(-3 \cdot \color{blue}{\left(-\sqrt{x}\right)}\right)} + 1 \]

       6. associate-*l* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(-\sqrt{x}\right)}} + 1 \]

       7. distribute-rgt-neg-out 95.0%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-\left(\sqrt{x} \cdot -3\right) \cdot \sqrt{x}}} + 1 \]

       8. *-commutative 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot -3\right)}} + 1 \]

       9. associate-*r* 95.0%

          \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot -3}} + 1 \]

       10. rem-square-sqrt 95.1%

           \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{x} \cdot -3} + 1 \]

       11. distribute-rgt-neg-in 95.1%

           \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot \left(--3\right)}} + 1 \]

       12. metadata-eval 95.1%

           \[\leadsto \frac{-0.3333333333333333}{x \cdot \color{blue}{3}} + 1 \]

   13. Simplified 95.1%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot 3}} + 1 \]

   if 8.89999999999999983e67 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around inf 88.8%

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

      1. *-commutative 88.8%

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

   9. Simplified 88.8%

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

       1. *-commutative 88.8%

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

       2. sqrt-div 88.8%

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

       3. metadata-eval 88.8%

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

       4. div-inv 88.7%

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

       5. associate-*l/ 88.7%

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

   11. Applied egg-rr 88.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq 8.9 \cdot 10^{+67}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 12: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 13: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y * N[Sqrt[N[(0.1111111111111111 / 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. 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. Step-by-step derivation

   1. *-un-lft-identity 99.7%

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

   2. sqrt-prod 99.7%

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

   3. metadata-eval 99.7%

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

   4. times-frac 99.7%

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

   5. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

   6. pow-flip 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

   7. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

8. Applied egg-rr 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]

9. Taylor expanded in x around 0 99.7%

   \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - {x}^{-0.5} \cdot \frac{y}{3} \]
10. Step-by-step derivation

    1. associate--l- 99.7%

       \[\leadsto \color{blue}{1 - \left(\frac{0.1111111111111111}{x} + {x}^{-0.5} \cdot \frac{y}{3}\right)} \]

    2. sub-neg 99.7%

       \[\leadsto \color{blue}{1 + \left(-\left(\frac{0.1111111111111111}{x} + {x}^{-0.5} \cdot \frac{y}{3}\right)\right)} \]

    3. *-commutative 99.7%

       \[\leadsto 1 + \left(-\left(\frac{0.1111111111111111}{x} + \color{blue}{\frac{y}{3} \cdot {x}^{-0.5}}\right)\right) \]

    4. div-inv 99.6%

       \[\leadsto 1 + \left(-\left(\frac{0.1111111111111111}{x} + \color{blue}{\left(y \cdot \frac{1}{3}\right)} \cdot {x}^{-0.5}\right)\right) \]

    5. metadata-eval 99.6%

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

    6. metadata-eval 99.6%

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

    7. associate-*l* 99.6%

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

    8. metadata-eval 99.6%

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

    9. metadata-eval 99.6%

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

    10. metadata-eval 99.6%

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

    11. sqrt-pow1 99.6%

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

    12. inv-pow 99.6%

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

    13. sqrt-prod 99.6%

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

    14. div-inv 99.7%

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

11. Applied egg-rr 99.7%

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

    1. distribute-neg-in 99.7%

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

    2. distribute-neg-frac 99.7%

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

    3. metadata-eval 99.7%

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

    4. associate-+r+ 99.7%

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

    5. sub-neg 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 14: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

4. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 99.7%

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

8. Final simplification 99.7%

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

Alternative 15: 62.0% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.024:\\ \;\;\;\;-0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.024) (* -0.1111111111111111 (/ 1.0 x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.024) {
		tmp = -0.1111111111111111 * (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.024d0) then
        tmp = (-0.1111111111111111d0) * (1.0d0 / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.024) {
		tmp = -0.1111111111111111 * (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.024:
		tmp = -0.1111111111111111 * (1.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.024)
		tmp = Float64(-0.1111111111111111 * Float64(1.0 / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.024)
		tmp = -0.1111111111111111 * (1.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.024], N[(-0.1111111111111111 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.024

   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 0 56.6%

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

      1. clear-num 56.5%

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

      2. associate-/r/ 56.6%

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

   9. Applied egg-rr 56.6%

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

   if 0.024 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 63.0%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.024:\\ \;\;\;\;-0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.0% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.024:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.024) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.024) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.024d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.024) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.024:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.024)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.024)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.024], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.024

   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 0 56.6%

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

   if 0.024 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.7%

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

      10. fma-neg 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      14. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      15. distribute-neg-frac 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      16. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      17. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 63.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.024:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.0% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.3333333333333333}{x \cdot 3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ -0.3333333333333333 (* x 3.0))))

C

double code(double x, double y) {
	return 1.0 + (-0.3333333333333333 / (x * 3.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-0.3333333333333333d0) / (x * 3.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-0.3333333333333333 / (x * 3.0));
}

Python

def code(x, y):
	return 1.0 + (-0.3333333333333333 / (x * 3.0))

Julia

function code(x, y)
	return Float64(1.0 + Float64(-0.3333333333333333 / Float64(x * 3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-0.3333333333333333 / (x * 3.0));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. +-commutative 99.7%

      \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

   4. associate-+r- 99.7%

      \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

   5. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. *-commutative 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   9. associate-*r/ 99.7%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fma-neg 99.7%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   11. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   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 60.9%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 60.9%

      \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

   2. metadata-eval 60.9%

      \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

   3. associate-*r/ 60.8%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

   4. metadata-eval 60.8%

      \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

   5. +-commutative 60.8%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

7. Simplified 60.8%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

      \[\leadsto \color{blue}{\sqrt{\frac{-0.1111111111111111}{x}} \cdot \sqrt{\frac{-0.1111111111111111}{x}}} + 1 \]

   2. sqrt-unprod 38.3%

      \[\leadsto \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}} + 1 \]

   3. frac-times 38.3%

      \[\leadsto \sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}} + 1 \]

   4. metadata-eval 38.3%

      \[\leadsto \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} + 1 \]

   5. metadata-eval 38.3%

      \[\leadsto \sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}} + 1 \]

   6. frac-times 38.3%

      \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}} + 1 \]

   7. sqrt-unprod 36.1%

      \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}} \cdot \sqrt{\frac{0.1111111111111111}{x}}} + 1 \]

   8. add-sqr-sqrt 36.1%

      \[\leadsto \color{blue}{\frac{0.1111111111111111}{x}} + 1 \]

   9. clear-num 36.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} + 1 \]

   10. div-inv 36.1%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} + 1 \]

   11. metadata-eval 36.1%

       \[\leadsto \frac{1}{x \cdot \color{blue}{9}} + 1 \]

   12. inv-pow 36.1%

       \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{-1}} + 1 \]

9. Applied egg-rr 36.1%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{-1}} + 1 \]
10. Step-by-step derivation

    1. inv-pow 36.1%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 9}} + 1 \]

    2. add-sqr-sqrt 36.1%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}}} + 1 \]

    3. associate-/r* 36.1%

       \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x \cdot 9}}}{\sqrt{x \cdot 9}}} + 1 \]

    4. metadata-eval 36.1%

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{x \cdot 9}}}{\sqrt{x \cdot 9}} + 1 \]

    5. sqrt-div 36.1%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x \cdot 9}}}}{\sqrt{x \cdot 9}} + 1 \]

    6. inv-pow 36.1%

       \[\leadsto \frac{\sqrt{\color{blue}{{\left(x \cdot 9\right)}^{-1}}}}{\sqrt{x \cdot 9}} + 1 \]

    7. inv-pow 36.1%

       \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{x \cdot 9}}}}{\sqrt{x \cdot 9}} + 1 \]

    8. metadata-eval 36.1%

       \[\leadsto \frac{\sqrt{\frac{1}{x \cdot \color{blue}{\frac{1}{0.1111111111111111}}}}}{\sqrt{x \cdot 9}} + 1 \]

    9. div-inv 36.1%

       \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\frac{x}{0.1111111111111111}}}}}{\sqrt{x \cdot 9}} + 1 \]

    10. clear-num 36.1%

        \[\leadsto \frac{\sqrt{\color{blue}{\frac{0.1111111111111111}{x}}}}{\sqrt{x \cdot 9}} + 1 \]

    11. add-sqr-sqrt 36.1%

        \[\leadsto \frac{\sqrt{\frac{0.1111111111111111}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

    12. metadata-eval 36.1%

        \[\leadsto \frac{\sqrt{\frac{\color{blue}{-0.3333333333333333 \cdot -0.3333333333333333}}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

    13. frac-times 36.1%

        \[\leadsto \frac{\sqrt{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot \frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

    14. sqrt-prod 0.0%

        \[\leadsto \frac{\color{blue}{\sqrt{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot \sqrt{\frac{-0.3333333333333333}{\sqrt{x}}}}}{\sqrt{x \cdot 9}} + 1 \]

    15. add-sqr-sqrt 60.8%

        \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}}}{\sqrt{x \cdot 9}} + 1 \]

    16. *-commutative 60.8%

        \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{9 \cdot x}}} + 1 \]

    17. sqrt-prod 60.8%

        \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{9} \cdot \sqrt{x}}} + 1 \]

    18. metadata-eval 60.8%

        \[\leadsto \frac{\frac{-0.3333333333333333}{\sqrt{x}}}{\color{blue}{3} \cdot \sqrt{x}} + 1 \]

11. Applied egg-rr 60.8%

    \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\sqrt{x}}}{3 \cdot \sqrt{x}}} + 1 \]
12. Step-by-step derivation

    1. associate-/l/ 60.8%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\left(3 \cdot \sqrt{x}\right) \cdot \sqrt{x}}} + 1 \]

    2. *-commutative 60.8%

       \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \left(3 \cdot \sqrt{x}\right)}} + 1 \]

    3. metadata-eval 60.8%

       \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(\color{blue}{\left(-3 \cdot -1\right)} \cdot \sqrt{x}\right)} + 1 \]

    4. associate-*r* 60.8%

       \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \color{blue}{\left(-3 \cdot \left(-1 \cdot \sqrt{x}\right)\right)}} + 1 \]

    5. neg-mul-1 60.8%

       \[\leadsto \frac{-0.3333333333333333}{\sqrt{x} \cdot \left(-3 \cdot \color{blue}{\left(-\sqrt{x}\right)}\right)} + 1 \]

    6. associate-*l* 60.8%

       \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\sqrt{x} \cdot -3\right) \cdot \left(-\sqrt{x}\right)}} + 1 \]

    7. distribute-rgt-neg-out 60.8%

       \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-\left(\sqrt{x} \cdot -3\right) \cdot \sqrt{x}}} + 1 \]

    8. *-commutative 60.8%

       \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot -3\right)}} + 1 \]

    9. associate-*r* 60.8%

       \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot -3}} + 1 \]

    10. rem-square-sqrt 60.9%

        \[\leadsto \frac{-0.3333333333333333}{-\color{blue}{x} \cdot -3} + 1 \]

    11. distribute-rgt-neg-in 60.9%

        \[\leadsto \frac{-0.3333333333333333}{\color{blue}{x \cdot \left(--3\right)}} + 1 \]

    12. metadata-eval 60.9%

        \[\leadsto \frac{-0.3333333333333333}{x \cdot \color{blue}{3}} + 1 \]

13. Simplified 60.9%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333}{x \cdot 3}} + 1 \]

14. Final simplification 60.9%

    \[\leadsto 1 + \frac{-0.3333333333333333}{x \cdot 3} \]
15. Add Preprocessing

Alternative 18: 63.0% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + 0.1111111111111111 \cdot \frac{-1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.1111111111111111 (/ -1.0 x))))

C

double code(double x, double y) {
	return 1.0 + (0.1111111111111111 * (-1.0 / x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.1111111111111111 * (-1.0 / x));
}

Python

def code(x, y):
	return 1.0 + (0.1111111111111111 * (-1.0 / x))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. +-commutative 99.7%

      \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

   4. associate-+r- 99.7%

      \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

   5. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. *-commutative 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   9. associate-*r/ 99.7%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fma-neg 99.7%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   11. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   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 60.9%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

6. Final simplification 60.9%

   \[\leadsto 1 + 0.1111111111111111 \cdot \frac{-1}{x} \]
7. Add Preprocessing

Alternative 19: 63.0% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ -0.1111111111111111 x)))

C

double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-0.1111111111111111d0) / x)
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

function code(x, y)
	return Float64(1.0 + Float64(-0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-0.1111111111111111 / x);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. +-commutative 99.7%

      \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

   4. associate-+r- 99.7%

      \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

   5. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. *-commutative 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   9. associate-*r/ 99.7%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fma-neg 99.7%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   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 60.9%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 60.9%

      \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

   2. metadata-eval 60.9%

      \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

   3. associate-*r/ 60.8%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

   4. metadata-eval 60.8%

      \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

   5. +-commutative 60.8%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

7. Simplified 60.8%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

8. Final simplification 60.8%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]
9. Add Preprocessing

Alternative 20: 32.5% 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. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. +-commutative 99.7%

      \[\leadsto \color{blue}{\frac{-y}{3 \cdot \sqrt{x}} + \left(1 - \frac{1}{x \cdot 9}\right)} \]

   4. associate-+r- 99.7%

      \[\leadsto \color{blue}{\left(\frac{-y}{3 \cdot \sqrt{x}} + 1\right) - \frac{1}{x \cdot 9}} \]

   5. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{-y}{3 \cdot \sqrt{x}}\right)} - \frac{1}{x \cdot 9} \]

   6. associate-+r- 99.7%

      \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

   7. neg-mul-1 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   8. *-commutative 99.7%

      \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot -1}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

   9. associate-*r/ 99.7%

      \[\leadsto 1 + \left(\color{blue}{y \cdot \frac{-1}{3 \cdot \sqrt{x}}} - \frac{1}{x \cdot 9}\right) \]

   10. fma-neg 99.7%

       \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(y, \frac{-1}{3 \cdot \sqrt{x}}, -\frac{1}{x \cdot 9}\right)} \]

   11. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, -\frac{1}{x \cdot 9}\right) \]

   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 x around inf 36.0%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 36.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - ((1.0 / x) / 9.0)) - (y / (3.0 * sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - ((1.0d0 / x) / 9.0d0)) - (y / (3.0d0 * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - ((1.0 / x) / 9.0)) - (y / (3.0 * Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - ((1.0 / x) / 9.0)) - (y / (3.0 * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(Float64(1.0 / x) / 9.0)) - Float64(y / Float64(3.0 * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - ((1.0 / x) / 9.0)) - (y / (3.0 * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(N[(1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024021 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))