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

Percentage Accurate: 99.7% → 99.7%

Time: 9.5s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Final simplification 99.7%

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

Alternative 2: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+36} \lor \neg \left(y \leq 3.5 \cdot 10^{+48}\right):\\ \;\;\;\;1 + \left(\frac{0.1111111111111111}{x} - \frac{\frac{y}{\sqrt{x}}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.7e+36) (not (<= y 3.5e+48)))
   (+ 1.0 (- (/ 0.1111111111111111 x) (/ (/ y (sqrt x)) 3.0)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.7e+36) || !(y <= 3.5e+48)) {
		tmp = 1.0 + ((0.1111111111111111 / x) - ((y / sqrt(x)) / 3.0));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3.7d+36)) .or. (.not. (y <= 3.5d+48))) then
        tmp = 1.0d0 + ((0.1111111111111111d0 / x) - ((y / sqrt(x)) / 3.0d0))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.7e+36) || !(y <= 3.5e+48)) {
		tmp = 1.0 + ((0.1111111111111111 / x) - ((y / Math.sqrt(x)) / 3.0));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.7e+36) or not (y <= 3.5e+48):
		tmp = 1.0 + ((0.1111111111111111 / x) - ((y / math.sqrt(x)) / 3.0))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.7e+36) || !(y <= 3.5e+48))
		tmp = Float64(1.0 + Float64(Float64(0.1111111111111111 / x) - Float64(Float64(y / sqrt(x)) / 3.0)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.7e+36) || ~((y <= 3.5e+48)))
		tmp = 1.0 + ((0.1111111111111111 / x) - ((y / sqrt(x)) / 3.0));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.7e+36], N[Not[LessEqual[y, 3.5e+48]], $MachinePrecision]], N[(1.0 + N[(N[(0.1111111111111111 / x), $MachinePrecision] - N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.70000000000000029e36 or 3.4999999999999997e48 < 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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.4%

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

      4. pow1/2 99.4%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

      5. pow-flip 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

      6. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]
   6. Step-by-step derivation

      1. associate-/l* 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5}}{\frac{3}{y}}} \]

   7. Simplified 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5}}{\frac{3}{y}}} \]

   8. Taylor expanded in x around 0 99.5%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{{x}^{-0.5}}{\frac{3}{y}} \]
   9. Step-by-step derivation

      1. div-inv 99.5%

         \[\leadsto \left(1 - \color{blue}{0.1111111111111111 \cdot \frac{1}{x}}\right) - \frac{{x}^{-0.5}}{\frac{3}{y}} \]

      2. cancel-sign-sub-inv 99.5%

         \[\leadsto \color{blue}{\left(1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}\right)} - \frac{{x}^{-0.5}}{\frac{3}{y}} \]

      3. metadata-eval 99.5%

         \[\leadsto \left(1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x}\right) - \frac{{x}^{-0.5}}{\frac{3}{y}} \]

      4. associate--l+ 99.5%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111 \cdot \frac{1}{x} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right)} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto 1 + \left(\color{blue}{\sqrt{-0.1111111111111111 \cdot \frac{1}{x}} \cdot \sqrt{-0.1111111111111111 \cdot \frac{1}{x}}} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right) \]

      6. sqrt-unprod 80.5%

         \[\leadsto 1 + \left(\color{blue}{\sqrt{\left(-0.1111111111111111 \cdot \frac{1}{x}\right) \cdot \left(-0.1111111111111111 \cdot \frac{1}{x}\right)}} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right) \]

      7. un-div-inv 80.5%

         \[\leadsto 1 + \left(\sqrt{\color{blue}{\frac{-0.1111111111111111}{x}} \cdot \left(-0.1111111111111111 \cdot \frac{1}{x}\right)} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right) \]

      8. un-div-inv 80.5%

         \[\leadsto 1 + \left(\sqrt{\frac{-0.1111111111111111}{x} \cdot \color{blue}{\frac{-0.1111111111111111}{x}}} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right) \]

      9. frac-times 80.5%

         \[\leadsto 1 + \left(\sqrt{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right) \]

      10. metadata-eval 80.5%

          \[\leadsto 1 + \left(\sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right) \]

      11. metadata-eval 80.5%

          \[\leadsto 1 + \left(\sqrt{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{x \cdot x}} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right) \]

      12. frac-times 80.5%

          \[\leadsto 1 + \left(\sqrt{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right) \]

      13. sqrt-unprod 90.0%

          \[\leadsto 1 + \left(\color{blue}{\sqrt{\frac{0.1111111111111111}{x}} \cdot \sqrt{\frac{0.1111111111111111}{x}}} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right) \]

      14. add-sqr-sqrt 90.0%

          \[\leadsto 1 + \left(\color{blue}{\frac{0.1111111111111111}{x}} - \frac{{x}^{-0.5}}{\frac{3}{y}}\right) \]

      15. associate-/r/ 90.0%

          \[\leadsto 1 + \left(\frac{0.1111111111111111}{x} - \color{blue}{\frac{{x}^{-0.5}}{3} \cdot y}\right) \]

      16. associate-*l/ 90.0%

          \[\leadsto 1 + \left(\frac{0.1111111111111111}{x} - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}}\right) \]

   10. Applied egg-rr 90.0%

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

   if -3.70000000000000029e36 < y < 3.4999999999999997e48

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r* 99.8%

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

      5. metadata-eval 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.8%

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

      1. metadata-eval 97.8%

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

      2. inv-pow 97.8%

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

      3. unpow-prod-down 97.8%

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

      4. *-commutative 97.8%

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

      5. inv-pow 97.8%

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

   6. Applied egg-rr 97.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+36} \lor \neg \left(y \leq 3.5 \cdot 10^{+48}\right):\\ \;\;\;\;1 + \left(\frac{0.1111111111111111}{x} - \frac{\frac{y}{\sqrt{x}}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 3: 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. 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%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

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(y * Float64(-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[(y * N[(-0.3333333333333333 / 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. 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%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. expm1-log1p-u 74.0%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\right)\right)} \]

   2. expm1-udef 74.0%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\right)} - 1\right)} \]

5. Applied egg-rr 74.0%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 74.0%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\right)\right)} \]

   2. expm1-log1p 99.6%

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

   3. associate-*r/ 99.5%

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

   4. associate-*l/ 99.6%

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

   5. *-commutative 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

   3. *-commutative 99.7%

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

   4. unpow-prod-down 99.7%

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

   5. metadata-eval 99.7%

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

   6. inv-pow 99.7%

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

   7. div-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. metadata-eval 99.7%

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

   10. sqrt-prod 99.7%

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

3. Applied egg-rr 99.7%

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

   1. unsub-neg 99.7%

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

   2. sub-neg 99.7%

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

   3. distribute-neg-frac 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.45 \cdot 10^{+36} \lor \neg \left(y \leq 4 \cdot 10^{+105}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.45e+36) (not (<= y 4e+105)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.45e+36) || !(y <= 4e+105)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.45d+36)) .or. (.not. (y <= 4d+105))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.45e+36) || !(y <= 4e+105)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.45e+36) or not (y <= 4e+105):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.45e+36) || !(y <= 4e+105))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.45e+36) || ~((y <= 4e+105)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.45e+36], N[Not[LessEqual[y, 4e+105]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.44999999999999999e36 or 3.9999999999999998e105 < 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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.4%

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

      4. pow1/2 99.4%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

      5. pow-flip 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

      6. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]

   4. Taylor expanded in y around inf 85.9%

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

      1. *-commutative 85.9%

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

   6. Simplified 85.9%

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

      1. expm1-log1p-u 33.6%

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

      2. expm1-udef 33.4%

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

      3. sqrt-div 33.4%

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

      4. metadata-eval 33.4%

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

      5. associate-*l/ 33.4%

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

      6. *-un-lft-identity 33.4%

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

   8. Applied egg-rr 33.4%

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

      1. expm1-def 33.6%

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

      2. expm1-log1p 85.8%

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

   10. Simplified 85.8%

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

   if -4.44999999999999999e36 < y < 3.9999999999999998e105

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r* 99.7%

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

      5. metadata-eval 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 96.1%

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

      1. metadata-eval 96.1%

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

      2. inv-pow 96.1%

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

      3. unpow-prod-down 96.2%

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

      4. *-commutative 96.2%

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

      5. inv-pow 96.2%

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

   6. Applied egg-rr 96.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.45 \cdot 10^{+36} \lor \neg \left(y \leq 4 \cdot 10^{+105}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 7: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.45 \cdot 10^{+36}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+105}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.45e+36)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 4e+105)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.45e+36) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 4e+105) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.45d+36)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 4d+105) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.45e+36) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 4e+105) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.45e+36:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 4e+105:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.45e+36)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 4e+105)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.45e+36)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 4e+105)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.45e+36], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+105], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.44999999999999999e36

   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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.5%

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

      4. pow1/2 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

      5. pow-flip 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

      6. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]

   4. Taylor expanded in y around inf 81.8%

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

      1. *-commutative 81.8%

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

   6. Simplified 81.8%

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

      1. expm1-log1p-u 0.6%

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

      2. expm1-udef 0.4%

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

      3. sqrt-div 0.4%

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

      4. metadata-eval 0.4%

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

      5. associate-*l/ 0.4%

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

      6. *-un-lft-identity 0.4%

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

   8. Applied egg-rr 0.4%

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

      1. expm1-def 0.6%

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

      2. expm1-log1p 81.7%

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

   10. Simplified 81.7%

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

   if -4.44999999999999999e36 < y < 3.9999999999999998e105

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r* 99.7%

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

      5. metadata-eval 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 96.1%

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

      1. metadata-eval 96.1%

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

      2. inv-pow 96.1%

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

      3. unpow-prod-down 96.2%

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

      4. *-commutative 96.2%

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

      5. inv-pow 96.2%

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

   6. Applied egg-rr 96.2%

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

   if 3.9999999999999998e105 < 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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.4%

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

      4. pow1/2 99.4%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

      5. pow-flip 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

      6. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]

   4. Taylor expanded in y around inf 92.3%

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

      1. *-commutative 92.3%

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

   6. Simplified 92.3%

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

      1. associate-*l* 92.1%

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

      2. sqrt-div 92.2%

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

      3. metadata-eval 92.2%

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

      4. associate-*l/ 92.3%

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

      5. *-un-lft-identity 92.3%

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

   8. Applied egg-rr 92.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.45 \cdot 10^{+36}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+105}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 8: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.45 \cdot 10^{+36}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+105}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.45e+36)
   (* (* y -0.3333333333333333) (pow x -0.5))
   (if (<= y 4e+105)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.45e+36) {
		tmp = (y * -0.3333333333333333) * pow(x, -0.5);
	} else if (y <= 4e+105) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.45d+36)) then
        tmp = (y * (-0.3333333333333333d0)) * (x ** (-0.5d0))
    else if (y <= 4d+105) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.45e+36) {
		tmp = (y * -0.3333333333333333) * Math.pow(x, -0.5);
	} else if (y <= 4e+105) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.45e+36:
		tmp = (y * -0.3333333333333333) * math.pow(x, -0.5)
	elif y <= 4e+105:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.45e+36)
		tmp = Float64(Float64(y * -0.3333333333333333) * (x ^ -0.5));
	elseif (y <= 4e+105)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.45e+36)
		tmp = (y * -0.3333333333333333) * (x ^ -0.5);
	elseif (y <= 4e+105)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.45e+36], N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+105], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.44999999999999999e36

   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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.5%

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

      4. pow1/2 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

      5. pow-flip 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

      6. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5} \cdot y}{3}} \]
   6. Step-by-step derivation

      1. associate-/l* 99.4%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5}}{\frac{3}{y}}} \]

   7. Simplified 99.4%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{{x}^{-0.5}}{\frac{3}{y}}} \]

   8. Taylor expanded in y around inf 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-*l* 81.7%

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

      3. unpow1/2 81.7%

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

      4. rem-exp-log 77.3%

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

      5. exp-neg 77.3%

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

      6. exp-prod 77.3%

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

      7. distribute-lft-neg-out 77.3%

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

      8. distribute-rgt-neg-in 77.3%

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

      9. metadata-eval 77.3%

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

      10. exp-to-pow 81.8%

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

   10. Simplified 81.8%

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

   if -4.44999999999999999e36 < y < 3.9999999999999998e105

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r* 99.7%

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

      5. metadata-eval 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 96.1%

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

      1. metadata-eval 96.1%

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

      2. inv-pow 96.1%

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

      3. unpow-prod-down 96.2%

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

      4. *-commutative 96.2%

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

      5. inv-pow 96.2%

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

   6. Applied egg-rr 96.2%

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

   if 3.9999999999999998e105 < 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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.4%

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

      4. pow1/2 99.4%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

      5. pow-flip 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

      6. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]

   4. Taylor expanded in y around inf 92.3%

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

      1. *-commutative 92.3%

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

   6. Simplified 92.3%

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

      1. associate-*l* 92.1%

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

      2. sqrt-div 92.2%

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

      3. metadata-eval 92.2%

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

      4. associate-*l/ 92.3%

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

      5. *-un-lft-identity 92.3%

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

   8. Applied egg-rr 92.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.45 \cdot 10^{+36}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+105}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 9: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.45 \cdot 10^{+36}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+105}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.45e+36)
   (* -0.3333333333333333 (* y (sqrt (/ 1.0 x))))
   (if (<= y 5.6e+105)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.45e+36) {
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	} else if (y <= 5.6e+105) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.45d+36)) then
        tmp = (-0.3333333333333333d0) * (y * sqrt((1.0d0 / x)))
    else if (y <= 5.6d+105) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.45e+36) {
		tmp = -0.3333333333333333 * (y * Math.sqrt((1.0 / x)));
	} else if (y <= 5.6e+105) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.45e+36:
		tmp = -0.3333333333333333 * (y * math.sqrt((1.0 / x)))
	elif y <= 5.6e+105:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.45e+36)
		tmp = Float64(-0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x))));
	elseif (y <= 5.6e+105)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.45e+36)
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	elseif (y <= 5.6e+105)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.45e+36], N[(-0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e+105], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.44999999999999999e36

   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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.5%

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

      4. pow1/2 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

      5. pow-flip 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

      6. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]

   4. Taylor expanded in y around inf 81.8%

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

      1. *-commutative 81.8%

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

   6. Simplified 81.8%

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

   if -4.44999999999999999e36 < y < 5.6000000000000003e105

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r* 99.7%

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

      5. metadata-eval 99.7%

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

      6. neg-mul-1 99.7%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 96.1%

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

      1. metadata-eval 96.1%

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

      2. inv-pow 96.1%

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

      3. unpow-prod-down 96.2%

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

      4. *-commutative 96.2%

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

      5. inv-pow 96.2%

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

   6. Applied egg-rr 96.2%

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

   if 5.6000000000000003e105 < 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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.4%

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

      4. pow1/2 99.4%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{y}{3} \]

      5. pow-flip 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{y}{3} \]

      6. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{-0.5}} \cdot \frac{y}{3} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]

   4. Taylor expanded in y around inf 92.3%

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

      1. *-commutative 92.3%

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

   6. Simplified 92.3%

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

      1. associate-*l* 92.1%

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

      2. sqrt-div 92.2%

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

      3. metadata-eval 92.2%

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

      4. associate-*l/ 92.3%

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

      5. *-un-lft-identity 92.3%

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

   8. Applied egg-rr 92.3%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.45 \cdot 10^{+36}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+105}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 10: 61.4% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-5}:\\ \;\;\;\;-0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.8e-5) (* -0.1111111111111111 (/ 1.0 x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.8e-5) {
		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 <= 7.8d-5) 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 <= 7.8e-5) {
		tmp = -0.1111111111111111 * (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.8e-5:
		tmp = -0.1111111111111111 * (1.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.8e-5)
		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 <= 7.8e-5)
		tmp = -0.1111111111111111 * (1.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.8e-5], N[(-0.1111111111111111 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 7.7999999999999999e-5

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. 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%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. neg-mul-1 99.4%

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

      7. times-frac 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 55.1%

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

      1. div-inv 55.2%

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

      2. inv-pow 55.2%

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

   6. Applied egg-rr 55.2%

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

   7. Taylor expanded in x around 0 55.2%

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

   if 7.7999999999999999e-5 < x

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r* 99.9%

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

      5. metadata-eval 99.9%

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

      6. neg-mul-1 99.9%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 69.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-5}:\\ \;\;\;\;-0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 62.5% 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%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 63.6%

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

5. Final simplification 63.6%

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

Alternative 12: 62.5% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 63.6%

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

   1. metadata-eval 63.6%

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

   2. inv-pow 63.6%

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

   3. unpow-prod-down 63.7%

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

   4. *-commutative 63.7%

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

   5. inv-pow 63.7%

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

6. Applied egg-rr 63.7%

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

7. Final simplification 63.7%

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

Alternative 13: 61.4% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.8e-5) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.8e-5) {
		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 <= 7.8d-5) 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 <= 7.8e-5) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.8e-5:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.8e-5)
		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 <= 7.8e-5)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.8e-5], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 7.7999999999999999e-5

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. 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%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. neg-mul-1 99.4%

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

      7. times-frac 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 55.1%

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

   if 7.7999999999999999e-5 < x

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r* 99.9%

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

      5. metadata-eval 99.9%

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

      6. neg-mul-1 99.9%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 69.9%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 62.5% 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%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around 0 63.6%

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

   1. cancel-sign-sub-inv 63.6%

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

   2. metadata-eval 63.6%

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

   3. associate-*r/ 63.6%

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

   4. metadata-eval 63.6%

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

6. Simplified 63.6%

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

7. Final simplification 63.6%

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

Alternative 15: 31.5% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. 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%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around inf 38.0%

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

5. Final simplification 38.0%

   \[\leadsto 1 \]

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 2023275 
(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)))))