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

Percentage Accurate: 99.7% → 99.7%

Time: 8.7s

Alternatives: 12

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

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

FPCore

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

C

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

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)) / 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-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 99.3%

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

   4. pow1/2 99.3%

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

3. Applied egg-rr 99.3%

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

   1. unpow1/2 99.3%

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

5. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   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%

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

7. Applied egg-rr 99.7%

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

   1. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot \frac{y}{3} \]

   2. sqrt-pow1 99.7%

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

   3. inv-pow 99.7%

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

   4. associate-*r/ 99.7%

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

   5. sqrt-div 99.7%

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

   6. metadata-eval 99.7%

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

   7. associate-*l/ 99.7%

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

   8. *-un-lft-identity 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 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. Final simplification 99.7%

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

Alternative 3: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+38}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+54}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e+38)
   (+ 1.0 (* (/ y (sqrt x)) -0.3333333333333333))
   (if (<= y 1.7e+54)
     (- 1.0 (pow (* x 9.0) -1.0))
     (+ 1.0 (* (pow x -0.5) (* y -0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+38) {
		tmp = 1.0 + ((y / sqrt(x)) * -0.3333333333333333);
	} else if (y <= 1.7e+54) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (pow(x, -0.5) * (y * -0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.5d+38)) then
        tmp = 1.0d0 + ((y / sqrt(x)) * (-0.3333333333333333d0))
    else if (y <= 1.7d+54) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 + ((x ** (-0.5d0)) * (y * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5e+38) {
		tmp = 1.0 + ((y / Math.sqrt(x)) * -0.3333333333333333);
	} else if (y <= 1.7e+54) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + (Math.pow(x, -0.5) * (y * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e+38:
		tmp = 1.0 + ((y / math.sqrt(x)) * -0.3333333333333333)
	elif y <= 1.7e+54:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 + (math.pow(x, -0.5) * (y * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e+38)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) * -0.3333333333333333));
	elseif (y <= 1.7e+54)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64((x ^ -0.5) * Float64(y * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e+38)
		tmp = 1.0 + ((y / sqrt(x)) * -0.3333333333333333);
	elseif (y <= 1.7e+54)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 + ((x ^ -0.5) * (y * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e+38], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+54], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.49999999999999985e38

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      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. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l* 93.7%

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

      3. *-commutative 93.7%

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

   6. Simplified 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*r* 93.7%

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

      3. sqrt-div 93.8%

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

      4. metadata-eval 93.8%

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

      5. associate-/r/ 93.8%

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

      6. associate-*l/ 93.7%

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

      7. metadata-eval 93.7%

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

   8. Applied egg-rr 93.7%

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

      1. div-inv 93.8%

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

      2. clear-num 93.8%

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

      3. *-commutative 93.8%

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

   10. Applied egg-rr 93.8%

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

   if -2.49999999999999985e38 < y < 1.7e54

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      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. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 45.4%

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

      3. pow1 45.4%

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

      4. frac-times 45.4%

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

      5. metadata-eval 45.4%

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

      6. metadata-eval 45.4%

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

      7. frac-times 45.4%

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

      8. pow-prod-down 45.4%

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

      9. pow-prod-up 45.4%

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

      10. clear-num 45.4%

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

      11. div-inv 45.4%

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

      12. metadata-eval 45.4%

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

      13. metadata-eval 45.4%

          \[\leadsto 1 + \sqrt{{\left(\frac{1}{x \cdot 9}\right)}^{\color{blue}{2}}} \]

      14. pow2 45.4%

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

      15. sqrt-unprod 45.5%

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

      16. add-sqr-sqrt 45.5%

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

      17. frac-2neg 45.5%

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

      18. metadata-eval 45.5%

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

      19. div-inv 45.5%

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

      20. distribute-rgt-neg-in 45.5%

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

      21. metadata-eval 45.5%

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

      22. metadata-eval 45.5%

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

      23. div-inv 45.5%

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

      24. clear-num 45.5%

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

   6. Applied egg-rr 45.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 79.8%

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

      3. pow1 79.8%

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

      4. frac-times 79.8%

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

      5. metadata-eval 79.8%

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

      6. metadata-eval 79.8%

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

      7. frac-times 79.8%

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

      8. pow-prod-down 79.8%

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

      9. pow-prod-up 79.8%

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

      10. clear-num 79.7%

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

      11. div-inv 79.8%

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

      12. metadata-eval 79.8%

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

      13. metadata-eval 79.8%

          \[\leadsto 1 + -1 \cdot \sqrt{{\left(\frac{1}{x \cdot 9}\right)}^{\color{blue}{2}}} \]

      14. pow2 79.8%

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

      15. sqrt-unprod 99.1%

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

      16. add-sqr-sqrt 99.3%

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

      17. inv-pow 99.3%

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

   8. Applied egg-rr 99.3%

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

   if 1.7e54 < 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. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. associate--r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-frac-neg 99.5%

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

      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. associate-*l/ 99.4%

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

      9. fma-neg 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l* 94.0%

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

      3. *-commutative 94.0%

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

   6. Simplified 94.0%

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

      1. expm1-log1p-u 92.6%

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

      2. expm1-udef 44.9%

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

      3. inv-pow 44.9%

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

      4. sqrt-pow1 44.9%

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

      5. metadata-eval 44.9%

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

   8. Applied egg-rr 44.9%

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

      1. expm1-def 92.6%

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

      2. expm1-log1p 94.0%

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

   10. Simplified 94.0%

       \[\leadsto 1 + \color{blue}{{x}^{-0.5}} \cdot \left(-0.3333333333333333 \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+38}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+54}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 4: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.82 \cdot 10^{+40} \lor \neg \left(y \leq 3.1 \cdot 10^{+54}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.82e+40) (not (<= y 3.1e+54)))
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.82e+40) || !(y <= 3.1e+54)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.82d+40)) .or. (.not. (y <= 3.1d+54))) then
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.82e+40) || !(y <= 3.1e+54)) {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.82e+40) or not (y <= 3.1e+54):
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.82e+40) || !(y <= 3.1e+54))
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.82e+40) || ~((y <= 3.1e+54)))
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.82e+40], N[Not[LessEqual[y, 3.1e+54]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.82e40 or 3.0999999999999999e54 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      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. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*l* 93.8%

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

      3. *-commutative 93.8%

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

   6. Simplified 93.8%

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

      1. *-commutative 93.8%

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

      2. associate-*r* 93.8%

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

      3. sqrt-div 93.8%

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

      4. metadata-eval 93.8%

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

      5. associate-/r/ 93.8%

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

      6. associate-*l/ 93.7%

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

      7. metadata-eval 93.7%

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

   8. Applied egg-rr 93.7%

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

      1. associate-/r/ 93.8%

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

   10. Applied egg-rr 93.8%

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

   if -1.82e40 < y < 3.0999999999999999e54

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      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. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.82 \cdot 10^{+40} \lor \neg \left(y \leq 3.1 \cdot 10^{+54}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 5: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+54}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+40)
   (+ 1.0 (* (/ y (sqrt x)) -0.3333333333333333))
   (if (<= y 3.2e+54)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+40) {
		tmp = 1.0 + ((y / sqrt(x)) * -0.3333333333333333);
	} else if (y <= 3.2e+54) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (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 <= (-1.2d+40)) then
        tmp = 1.0d0 + ((y / sqrt(x)) * (-0.3333333333333333d0))
    else if (y <= 3.2d+54) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+40) {
		tmp = 1.0 + ((y / Math.sqrt(x)) * -0.3333333333333333);
	} else if (y <= 3.2e+54) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.2e+40:
		tmp = 1.0 + ((y / math.sqrt(x)) * -0.3333333333333333)
	elif y <= 3.2e+54:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+40)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) * -0.3333333333333333));
	elseif (y <= 3.2e+54)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e+40)
		tmp = 1.0 + ((y / sqrt(x)) * -0.3333333333333333);
	elseif (y <= 3.2e+54)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.2e+40], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+54], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e40

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      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. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l* 93.7%

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

      3. *-commutative 93.7%

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

   6. Simplified 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*r* 93.7%

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

      3. sqrt-div 93.8%

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

      4. metadata-eval 93.8%

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

      5. associate-/r/ 93.8%

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

      6. associate-*l/ 93.7%

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

      7. metadata-eval 93.7%

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

   8. Applied egg-rr 93.7%

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

      1. div-inv 93.8%

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

      2. clear-num 93.8%

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

      3. *-commutative 93.8%

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

   10. Applied egg-rr 93.8%

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

   if -1.2e40 < y < 3.2e54

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      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. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.3%

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

   if 3.2e54 < 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. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. associate--r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-frac-neg 99.5%

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

      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. associate-*l/ 99.4%

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

      9. fma-neg 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l* 94.0%

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

      3. *-commutative 94.0%

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

   6. Simplified 94.0%

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

      1. *-commutative 94.0%

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

      2. associate-*r* 93.9%

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

      3. sqrt-div 93.9%

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

      4. metadata-eval 93.9%

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

      5. associate-/r/ 93.9%

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

      6. associate-*l/ 93.8%

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

      7. metadata-eval 93.8%

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

   8. Applied egg-rr 93.8%

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

      1. associate-/r/ 93.9%

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

   10. Applied egg-rr 93.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+54}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 6: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.96 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+54}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.96e+39)
   (+ 1.0 (* (/ y (sqrt x)) -0.3333333333333333))
   (if (<= y 1.22e+54)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.96e+39) {
		tmp = 1.0 + ((y / sqrt(x)) * -0.3333333333333333);
	} else if (y <= 1.22e+54) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + ((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 <= (-1.96d+39)) then
        tmp = 1.0d0 + ((y / sqrt(x)) * (-0.3333333333333333d0))
    else if (y <= 1.22d+54) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.96e+39) {
		tmp = 1.0 + ((y / Math.sqrt(x)) * -0.3333333333333333);
	} else if (y <= 1.22e+54) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.96e+39:
		tmp = 1.0 + ((y / math.sqrt(x)) * -0.3333333333333333)
	elif y <= 1.22e+54:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.96e+39)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) * -0.3333333333333333));
	elseif (y <= 1.22e+54)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.96e+39)
		tmp = 1.0 + ((y / sqrt(x)) * -0.3333333333333333);
	elseif (y <= 1.22e+54)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.96e+39], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.22e+54], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.96000000000000001e39

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      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. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l* 93.7%

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

      3. *-commutative 93.7%

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

   6. Simplified 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*r* 93.7%

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

      3. sqrt-div 93.8%

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

      4. metadata-eval 93.8%

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

      5. associate-/r/ 93.8%

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

      6. associate-*l/ 93.7%

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

      7. metadata-eval 93.7%

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

   8. Applied egg-rr 93.7%

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

      1. div-inv 93.8%

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

      2. clear-num 93.8%

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

      3. *-commutative 93.8%

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

   10. Applied egg-rr 93.8%

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

   if -1.96000000000000001e39 < y < 1.22e54

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      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. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.3%

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

   if 1.22e54 < 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. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. associate--r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-frac-neg 99.5%

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

      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. associate-*l/ 99.4%

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

      9. fma-neg 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l* 94.0%

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

      3. *-commutative 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in y around 0 93.9%

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

      1. unpow1/2 93.9%

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

      2. sqr-pow 93.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left({\left(\frac{1}{x}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{1}{x}\right)}^{\left(\frac{0.5}{2}\right)}\right)} \cdot y\right) \]

      3. sqr-pow 93.9%

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

      4. unpow-1 93.9%

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

      5. exp-to-pow 89.4%

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

      6. *-commutative 89.4%

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

      7. neg-mul-1 89.4%

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

      8. exp-prod 89.4%

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

      9. distribute-lft-neg-out 89.4%

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

      10. exp-neg 89.4%

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

      11. exp-to-pow 93.9%

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

      12. unpow1/2 93.9%

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

      13. associate-*l/ 93.9%

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

      14. *-lft-identity 93.9%

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

      15. associate-*r/ 94.0%

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

   9. Simplified 94.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.96 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+54}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 7: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+54}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.4e+40)
   (+ 1.0 (* (/ y (sqrt x)) -0.3333333333333333))
   (if (<= y 1.7e+54)
     (- 1.0 (pow (* x 9.0) -1.0))
     (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.4e+40) {
		tmp = 1.0 + ((y / sqrt(x)) * -0.3333333333333333);
	} else if (y <= 1.7e+54) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + ((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 <= (-7.4d+40)) then
        tmp = 1.0d0 + ((y / sqrt(x)) * (-0.3333333333333333d0))
    else if (y <= 1.7d+54) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.4e+40) {
		tmp = 1.0 + ((y / Math.sqrt(x)) * -0.3333333333333333);
	} else if (y <= 1.7e+54) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.4e+40:
		tmp = 1.0 + ((y / math.sqrt(x)) * -0.3333333333333333)
	elif y <= 1.7e+54:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.4e+40)
		tmp = Float64(1.0 + Float64(Float64(y / sqrt(x)) * -0.3333333333333333));
	elseif (y <= 1.7e+54)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.4e+40)
		tmp = 1.0 + ((y / sqrt(x)) * -0.3333333333333333);
	elseif (y <= 1.7e+54)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.4e+40], N[(1.0 + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+54], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.4e40

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      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. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l* 93.7%

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

      3. *-commutative 93.7%

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

   6. Simplified 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*r* 93.7%

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

      3. sqrt-div 93.8%

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

      4. metadata-eval 93.8%

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

      5. associate-/r/ 93.8%

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

      6. associate-*l/ 93.7%

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

      7. metadata-eval 93.7%

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

   8. Applied egg-rr 93.7%

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

      1. div-inv 93.8%

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

      2. clear-num 93.8%

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

      3. *-commutative 93.8%

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

   10. Applied egg-rr 93.8%

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

   if -7.4e40 < y < 1.7e54

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      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. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 45.4%

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

      3. pow1 45.4%

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

      4. frac-times 45.4%

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

      5. metadata-eval 45.4%

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

      6. metadata-eval 45.4%

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

      7. frac-times 45.4%

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

      8. pow-prod-down 45.4%

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

      9. pow-prod-up 45.4%

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

      10. clear-num 45.4%

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

      11. div-inv 45.4%

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

      12. metadata-eval 45.4%

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

      13. metadata-eval 45.4%

          \[\leadsto 1 + \sqrt{{\left(\frac{1}{x \cdot 9}\right)}^{\color{blue}{2}}} \]

      14. pow2 45.4%

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

      15. sqrt-unprod 45.5%

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

      16. add-sqr-sqrt 45.5%

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

      17. frac-2neg 45.5%

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

      18. metadata-eval 45.5%

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

      19. div-inv 45.5%

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

      20. distribute-rgt-neg-in 45.5%

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

      21. metadata-eval 45.5%

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

      22. metadata-eval 45.5%

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

      23. div-inv 45.5%

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

      24. clear-num 45.5%

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

   6. Applied egg-rr 45.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 79.8%

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

      3. pow1 79.8%

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

      4. frac-times 79.8%

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

      5. metadata-eval 79.8%

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

      6. metadata-eval 79.8%

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

      7. frac-times 79.8%

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

      8. pow-prod-down 79.8%

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

      9. pow-prod-up 79.8%

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

      10. clear-num 79.7%

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

      11. div-inv 79.8%

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

      12. metadata-eval 79.8%

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

      13. metadata-eval 79.8%

          \[\leadsto 1 + -1 \cdot \sqrt{{\left(\frac{1}{x \cdot 9}\right)}^{\color{blue}{2}}} \]

      14. pow2 79.8%

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

      15. sqrt-unprod 99.1%

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

      16. add-sqr-sqrt 99.3%

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

      17. inv-pow 99.3%

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

   8. Applied egg-rr 99.3%

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

   if 1.7e54 < 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. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. associate--r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-frac-neg 99.5%

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

      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. associate-*l/ 99.4%

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

      9. fma-neg 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l* 94.0%

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

      3. *-commutative 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in y around 0 93.9%

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

      1. unpow1/2 93.9%

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

      2. sqr-pow 93.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(\color{blue}{\left({\left(\frac{1}{x}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{1}{x}\right)}^{\left(\frac{0.5}{2}\right)}\right)} \cdot y\right) \]

      3. sqr-pow 93.9%

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

      4. unpow-1 93.9%

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

      5. exp-to-pow 89.4%

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

      6. *-commutative 89.4%

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

      7. neg-mul-1 89.4%

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

      8. exp-prod 89.4%

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

      9. distribute-lft-neg-out 89.4%

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

      10. exp-neg 89.4%

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

      11. exp-to-pow 93.9%

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

      12. unpow1/2 93.9%

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

      13. associate-*l/ 93.9%

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

      14. *-lft-identity 93.9%

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

      15. associate-*r/ 94.0%

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

   9. Simplified 94.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+40}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+54}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) * (-0.3333333333333333d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 9: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[(N[Sqrt[x], $MachinePrecision] * 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. Taylor expanded in x around 0 99.7%

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

3. Final simplification 99.7%

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

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - ((y / sqrt(x)) / 3.0);
}

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)) / 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-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 99.3%

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

   4. pow1/2 99.3%

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

3. Applied egg-rr 99.3%

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

   1. unpow1/2 99.3%

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

5. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   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%

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

7. Applied egg-rr 99.7%

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

   1. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - {x}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot \frac{y}{3} \]

   2. sqrt-pow1 99.7%

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

   3. inv-pow 99.7%

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

   4. associate-*r/ 99.7%

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

   5. sqrt-div 99.7%

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

   6. metadata-eval 99.7%

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

   7. associate-*l/ 99.7%

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

   8. *-un-lft-identity 99.7%

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in x around 0 99.7%

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

11. Final simplification 99.7%

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

Alternative 11: 62.8% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--r+ 99.7%

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

   4. sub-neg 99.7%

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

   5. distribute-frac-neg 99.7%

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

   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. associate-*l/ 99.7%

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

   9. fma-neg 99.7%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

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

   12. distribute-neg-frac 99.6%

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

   13. metadata-eval 99.6%

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

   14. *-commutative 99.6%

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

   15. associate-/r* 99.6%

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

   16. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 64.0%

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

5. Final simplification 64.0%

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

Alternative 12: 31.5% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--r+ 99.7%

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

   4. sub-neg 99.7%

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

   5. distribute-frac-neg 99.7%

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

   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. associate-*l/ 99.7%

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

   9. fma-neg 99.7%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

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

   12. distribute-neg-frac 99.6%

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

   13. metadata-eval 99.6%

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

   14. *-commutative 99.6%

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

   15. associate-/r* 99.6%

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

   16. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in y around inf 66.7%

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

   1. *-commutative 66.7%

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

   2. associate-*l* 66.7%

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

   3. *-commutative 66.7%

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

6. Simplified 66.7%

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

7. Taylor expanded in x around inf 31.0%

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

8. Final simplification 31.0%

   \[\leadsto 1 \]

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 2023333 
(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)))))