Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Percentage Accurate: 78.1% → 99.9%

Time: 13.2s

Alternatives: 9

Speedup: 12.3×

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Initial Program: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{e^{x \cdot \mathsf{log1p}\left(\frac{y}{x}\right)}}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) (exp (* x (log1p (/ y x)))))))
   (if (<= x -3.5e-15) t_0 (if (<= x 2e-36) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / x) / exp((x * log1p((y / x))));
	double tmp;
	if (x <= -3.5e-15) {
		tmp = t_0;
	} else if (x <= 2e-36) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / x) / Math.exp((x * Math.log1p((y / x))));
	double tmp;
	if (x <= -3.5e-15) {
		tmp = t_0;
	} else if (x <= 2e-36) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / x) / math.exp((x * math.log1p((y / x))))
	tmp = 0
	if x <= -3.5e-15:
		tmp = t_0
	elif x <= 2e-36:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / x) / exp(Float64(x * log1p(Float64(y / x)))))
	tmp = 0.0
	if (x <= -3.5e-15)
		tmp = t_0;
	elseif (x <= 2e-36)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / N[Exp[N[(x * N[Log[1 + N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e-15], t$95$0, If[LessEqual[x, 2e-36], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5000000000000001e-15 or 1.9999999999999999e-36 < x

   1. Initial program 73.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 73.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 73.6%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{{\left(\frac{x}{x + y}\right)}^{x}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{1}}{{\left(\frac{x}{x + y}\right)}^{x}}\right)\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\left(-1 \cdot \color{blue}{x}\right)}\right)\right) \]

      8. pow-unpow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left({\left(\frac{x}{x + y}\right)}^{-1}\right)}^{\color{blue}{x}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{1}{\frac{x}{x + y}}\right)}^{x}\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x + y}{x}\right)}^{x}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\left(\frac{x + y}{x}\right), \color{blue}{x}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + y\right), x\right), x\right)\right) \]

      13. +-lowering-+.f64 73.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right), x\right)\right) \]

   6. Applied egg-rr 73.6%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\color{blue}{\left(1 + \frac{y}{x}\right)}, x\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{y}{x}\right)\right), x\right)\right) \]

      2. /-lowering-/.f64 73.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, x\right)\right), x\right)\right) \]

   9. Simplified 73.6%

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

       1. pow-to-exp N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(e^{\log \left(1 + \frac{y}{x}\right) \cdot x}\right)\right) \]

       2. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{exp.f64}\left(\left(\log \left(1 + \frac{y}{x}\right) \cdot x\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{exp.f64}\left(\left(x \cdot \log \left(1 + \frac{y}{x}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \log \left(1 + \frac{y}{x}\right)\right)\right)\right) \]

       5. log1p-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\mathsf{log1p}\left(\frac{y}{x}\right)\right)\right)\right)\right) \]

       6. log1p-lowering-log1p.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log1p.f64}\left(\left(\frac{y}{x}\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 100.0%

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

   if -3.5000000000000001e-15 < x < 1.9999999999999999e-36

   1. Initial program 79.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 79.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 79.5%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{0 - y}}{x}\\ \mathbf{if}\;x \leq -640:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- 0.0 y)) x)))
   (if (<= x -640.0) t_0 (if (<= x 5.5e-7) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp((0.0 - y)) / x;
	double tmp;
	if (x <= -640.0) {
		tmp = t_0;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((0.0d0 - y)) / x
    if (x <= (-640.0d0)) then
        tmp = t_0
    else if (x <= 5.5d-7) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((0.0 - y)) / x;
	double tmp;
	if (x <= -640.0) {
		tmp = t_0;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((0.0 - y)) / x
	tmp = 0
	if x <= -640.0:
		tmp = t_0
	elif x <= 5.5e-7:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(0.0 - y)) / x)
	tmp = 0.0
	if (x <= -640.0)
		tmp = t_0;
	elseif (x <= 5.5e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((0.0 - y)) / x;
	tmp = 0.0;
	if (x <= -640.0)
		tmp = t_0;
	elseif (x <= 5.5e-7)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(0.0 - y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -640.0], t$95$0, If[LessEqual[x, 5.5e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -640 or 5.5000000000000003e-7 < x

   1. Initial program 71.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 71.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 99.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f64 99.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(y\right)\right), x\right) \]

   9. Applied egg-rr 99.7%

      \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   if -640 < x < 5.5000000000000003e-7

   1. Initial program 81.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 81.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   7. Simplified 98.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -640:\\ \;\;\;\;\frac{e^{0 - y}}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{0 - y}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{1}{x}}{1 + y \cdot \left(1 + y \cdot \left(0.5 + \frac{-0.5}{x}\right)\right)}\\ \mathbf{elif}\;x \leq -640:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 - y \cdot 0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.6e+203)
   (/ (/ 1.0 x) (+ 1.0 (* y (+ 1.0 (* y (+ 0.5 (/ -0.5 x)))))))
   (if (<= x -640.0)
     (/ (+ 1.0 (* y (+ (* y (- 0.5 (* y 0.16666666666666666))) -1.0))) x)
     (if (<= x 5.5e-7)
       (/ 1.0 x)
       (/
        1.0
        (*
         x
         (+ 1.0 (* y (+ 1.0 (* y (+ 0.5 (* y 0.16666666666666666))))))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.6e+203) {
		tmp = (1.0 / x) / (1.0 + (y * (1.0 + (y * (0.5 + (-0.5 / x))))));
	} else if (x <= -640.0) {
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.6d+203)) then
        tmp = (1.0d0 / x) / (1.0d0 + (y * (1.0d0 + (y * (0.5d0 + ((-0.5d0) / x))))))
    else if (x <= (-640.0d0)) then
        tmp = (1.0d0 + (y * ((y * (0.5d0 - (y * 0.16666666666666666d0))) + (-1.0d0)))) / x
    else if (x <= 5.5d-7) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * (1.0d0 + (y * (1.0d0 + (y * (0.5d0 + (y * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.6e+203) {
		tmp = (1.0 / x) / (1.0 + (y * (1.0 + (y * (0.5 + (-0.5 / x))))));
	} else if (x <= -640.0) {
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.6e+203:
		tmp = (1.0 / x) / (1.0 + (y * (1.0 + (y * (0.5 + (-0.5 / x))))))
	elif x <= -640.0:
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x
	elif x <= 5.5e-7:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.6e+203)
		tmp = Float64(Float64(1.0 / x) / Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * Float64(0.5 + Float64(-0.5 / x)))))));
	elseif (x <= -640.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(0.5 - Float64(y * 0.16666666666666666))) + -1.0))) / x);
	elseif (x <= 5.5e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * Float64(0.5 + Float64(y * 0.16666666666666666))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.6e+203)
		tmp = (1.0 / x) / (1.0 + (y * (1.0 + (y * (0.5 + (-0.5 / x))))));
	elseif (x <= -640.0)
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x;
	elseif (x <= 5.5e-7)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.6e+203], N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + N[(y * N[(1.0 + N[(y * N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -640.0], N[(N[(1.0 + N[(y * N[(N[(y * N[(0.5 - N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.5e-7], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(1.0 + N[(y * N[(1.0 + N[(y * N[(0.5 + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.60000000000000047e203

   1. Initial program 50.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 50.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 50.8%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{{\left(\frac{x}{x + y}\right)}^{x}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{1}}{{\left(\frac{x}{x + y}\right)}^{x}}\right)\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\left(-1 \cdot \color{blue}{x}\right)}\right)\right) \]

      8. pow-unpow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left({\left(\frac{x}{x + y}\right)}^{-1}\right)}^{\color{blue}{x}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{1}{\frac{x}{x + y}}\right)}^{x}\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x + y}{x}\right)}^{x}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\left(\frac{x + y}{x}\right), \color{blue}{x}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + y\right), x\right), x\right)\right) \]

      13. +-lowering-+.f64 50.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right), x\right)\right) \]

   6. Applied egg-rr 50.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \color{blue}{\left(1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{x}}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 78.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{x}\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 78.6%

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

   if -7.60000000000000047e203 < x < -640

   1. Initial program 78.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 78.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y\right) - 1\right)\right), x\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) - 1\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) - 1\right)\right)\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) - 1\right)\right)\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + -1\right)\right)\right), x\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} - \frac{1}{6} \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot y\right)\right)\right), -1\right)\right)\right), x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\frac{1}{2}, \left(y \cdot \frac{1}{6}\right)\right)\right), -1\right)\right)\right), x\right) \]

      15. *-lowering-*.f64 76.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right), -1\right)\right)\right), x\right) \]

   10. Simplified 76.6%

       \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(0.5 - y \cdot 0.16666666666666666\right) + -1\right)}}{x} \]

   if -640 < x < 5.5000000000000003e-7

   1. Initial program 81.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 81.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   7. Simplified 98.8%

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

   if 5.5000000000000003e-7 < x

   1. Initial program 74.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 74.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 74.2%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{{\left(\frac{x}{x + y}\right)}^{x}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{1}}{{\left(\frac{x}{x + y}\right)}^{x}}\right)\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\left(-1 \cdot \color{blue}{x}\right)}\right)\right) \]

      8. pow-unpow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left({\left(\frac{x}{x + y}\right)}^{-1}\right)}^{\color{blue}{x}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{1}{\frac{x}{x + y}}\right)}^{x}\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x + y}{x}\right)}^{x}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\left(\frac{x + y}{x}\right), \color{blue}{x}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + y\right), x\right), x\right)\right) \]

      13. +-lowering-+.f64 74.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right), x\right)\right) \]

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \color{blue}{\left(1 + y \cdot \left(1 + y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(1 + y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} + \color{blue}{\left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 83.1%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{1 + y \cdot \left(1 + y \cdot \left(0.5 + \left(y \cdot \left(\frac{0.3333333333333333}{x \cdot x} + \left(0.16666666666666666 + \frac{-0.5}{x}\right)\right) + \frac{-0.5}{x}\right)\right)\right)}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 83.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 83.5%

       \[\leadsto \color{blue}{\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -640:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 - y \cdot 0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          1.0
          (*
           x
           (+ 1.0 (* y (+ 1.0 (* y (+ 0.5 (* y 0.16666666666666666))))))))))
   (if (<= x -4.2e+203)
     t_0
     (if (<= x -640.0)
       (/ (+ 1.0 (* y (+ (* y (- 0.5 (* y 0.16666666666666666))) -1.0))) x)
       (if (<= x 5.5e-7) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	double tmp;
	if (x <= -4.2e+203) {
		tmp = t_0;
	} else if (x <= -640.0) {
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (x * (1.0d0 + (y * (1.0d0 + (y * (0.5d0 + (y * 0.16666666666666666d0)))))))
    if (x <= (-4.2d+203)) then
        tmp = t_0
    else if (x <= (-640.0d0)) then
        tmp = (1.0d0 + (y * ((y * (0.5d0 - (y * 0.16666666666666666d0))) + (-1.0d0)))) / x
    else if (x <= 5.5d-7) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	double tmp;
	if (x <= -4.2e+203) {
		tmp = t_0;
	} else if (x <= -640.0) {
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))))
	tmp = 0
	if x <= -4.2e+203:
		tmp = t_0
	elif x <= -640.0:
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x
	elif x <= 5.5e-7:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x * Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * Float64(0.5 + Float64(y * 0.16666666666666666))))))))
	tmp = 0.0
	if (x <= -4.2e+203)
		tmp = t_0;
	elseif (x <= -640.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(0.5 - Float64(y * 0.16666666666666666))) + -1.0))) / x);
	elseif (x <= 5.5e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (x * (1.0 + (y * (1.0 + (y * (0.5 + (y * 0.16666666666666666)))))));
	tmp = 0.0;
	if (x <= -4.2e+203)
		tmp = t_0;
	elseif (x <= -640.0)
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x;
	elseif (x <= 5.5e-7)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(1.0 + N[(y * N[(1.0 + N[(y * N[(0.5 + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.2e+203], t$95$0, If[LessEqual[x, -640.0], N[(N[(1.0 + N[(y * N[(N[(y * N[(0.5 - N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.5e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.19999999999999967e203 or 5.5000000000000003e-7 < x

   1. Initial program 69.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 69.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 69.8%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{{\left(\frac{x}{x + y}\right)}^{x}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{1}}{{\left(\frac{x}{x + y}\right)}^{x}}\right)\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\left(-1 \cdot \color{blue}{x}\right)}\right)\right) \]

      8. pow-unpow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left({\left(\frac{x}{x + y}\right)}^{-1}\right)}^{\color{blue}{x}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{1}{\frac{x}{x + y}}\right)}^{x}\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x + y}{x}\right)}^{x}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\left(\frac{x + y}{x}\right), \color{blue}{x}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + y\right), x\right), x\right)\right) \]

      13. +-lowering-+.f64 69.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right), x\right)\right) \]

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \color{blue}{\left(1 + y \cdot \left(1 + y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(1 + y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right)\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} + \color{blue}{\left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 82.3%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{1 + y \cdot \left(1 + y \cdot \left(0.5 + \left(y \cdot \left(\frac{0.3333333333333333}{x \cdot x} + \left(0.16666666666666666 + \frac{-0.5}{x}\right)\right) + \frac{-0.5}{x}\right)\right)\right)}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)}} \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 82.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 82.6%

       \[\leadsto \color{blue}{\frac{1}{x \cdot \left(1 + y \cdot \left(1 + y \cdot \left(0.5 + y \cdot 0.16666666666666666\right)\right)\right)}} \]

   if -4.19999999999999967e203 < x < -640

   1. Initial program 78.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 78.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y\right) - 1\right)\right), x\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) - 1\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) - 1\right)\right)\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) - 1\right)\right)\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + -1\right)\right)\right), x\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} - \frac{1}{6} \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot y\right)\right)\right), -1\right)\right)\right), x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\frac{1}{2}, \left(y \cdot \frac{1}{6}\right)\right)\right), -1\right)\right)\right), x\right) \]

      15. *-lowering-*.f64 76.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right), -1\right)\right)\right), x\right) \]

   10. Simplified 76.6%

       \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(0.5 - y \cdot 0.16666666666666666\right) + -1\right)}}{x} \]

   if -640 < x < 5.5000000000000003e-7

   1. Initial program 81.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 81.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{1 + y}\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -640:\\ \;\;\;\;\frac{1}{x} + y \cdot \frac{y \cdot \left(y \cdot -0.16666666666666666\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) (+ 1.0 y))))
   (if (<= x -4.4e+203)
     t_0
     (if (<= x -640.0)
       (+ (/ 1.0 x) (* y (/ (* y (* y -0.16666666666666666)) x)))
       (if (<= x 5.5e-7) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 / x) / (1.0 + y);
	double tmp;
	if (x <= -4.4e+203) {
		tmp = t_0;
	} else if (x <= -640.0) {
		tmp = (1.0 / x) + (y * ((y * (y * -0.16666666666666666)) / x));
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / (1.0d0 + y)
    if (x <= (-4.4d+203)) then
        tmp = t_0
    else if (x <= (-640.0d0)) then
        tmp = (1.0d0 / x) + (y * ((y * (y * (-0.16666666666666666d0))) / x))
    else if (x <= 5.5d-7) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / x) / (1.0 + y);
	double tmp;
	if (x <= -4.4e+203) {
		tmp = t_0;
	} else if (x <= -640.0) {
		tmp = (1.0 / x) + (y * ((y * (y * -0.16666666666666666)) / x));
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / x) / (1.0 + y)
	tmp = 0
	if x <= -4.4e+203:
		tmp = t_0
	elif x <= -640.0:
		tmp = (1.0 / x) + (y * ((y * (y * -0.16666666666666666)) / x))
	elif x <= 5.5e-7:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / x) / Float64(1.0 + y))
	tmp = 0.0
	if (x <= -4.4e+203)
		tmp = t_0;
	elseif (x <= -640.0)
		tmp = Float64(Float64(1.0 / x) + Float64(y * Float64(Float64(y * Float64(y * -0.16666666666666666)) / x)));
	elseif (x <= 5.5e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / x) / (1.0 + y);
	tmp = 0.0;
	if (x <= -4.4e+203)
		tmp = t_0;
	elseif (x <= -640.0)
		tmp = (1.0 / x) + (y * ((y * (y * -0.16666666666666666)) / x));
	elseif (x <= 5.5e-7)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e+203], t$95$0, If[LessEqual[x, -640.0], N[(N[(1.0 / x), $MachinePrecision] + N[(y * N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.40000000000000009e203 or 5.5000000000000003e-7 < x

   1. Initial program 69.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 69.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 69.8%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{{\left(\frac{x}{x + y}\right)}^{x}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{1}}{{\left(\frac{x}{x + y}\right)}^{x}}\right)\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\left(-1 \cdot \color{blue}{x}\right)}\right)\right) \]

      8. pow-unpow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left({\left(\frac{x}{x + y}\right)}^{-1}\right)}^{\color{blue}{x}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{1}{\frac{x}{x + y}}\right)}^{x}\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x + y}{x}\right)}^{x}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\left(\frac{x + y}{x}\right), \color{blue}{x}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + y\right), x\right), x\right)\right) \]

      13. +-lowering-+.f64 69.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right), x\right)\right) \]

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \color{blue}{\left(1 + y\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 71.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \color{blue}{y}\right)\right) \]

   9. Simplified 71.9%

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

   if -4.40000000000000009e203 < x < -640

   1. Initial program 78.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 78.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(-1 \cdot \frac{y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}{x} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) - \frac{1}{x}\right) + \frac{1}{x}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{x} + \color{blue}{y \cdot \left(y \cdot \left(-1 \cdot \frac{y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}{x} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) - \frac{1}{x}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(y \cdot \left(y \cdot \left(-1 \cdot \frac{y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}{x} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) - \frac{1}{x}\right)\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{y} \cdot \left(y \cdot \left(-1 \cdot \frac{y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}{x} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) - \frac{1}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(-1 \cdot \frac{y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}{x} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) - \frac{1}{x}\right)}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(-1 \cdot \frac{y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}{x} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(-1 \cdot \frac{y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}{x} + \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right)\right)\right) \]

   7. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{1}{x} + y \cdot \left(y \cdot \left(\left(\frac{0.5}{x} + \frac{0.5}{x \cdot x}\right) - \left(\frac{0.5}{x} + \left(0.16666666666666666 + \frac{0.3333333333333333}{x \cdot x}\right)\right) \cdot \frac{y}{x}\right) + \frac{-1}{x}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\frac{1}{2} - \frac{1}{6} \cdot y}{x}\right)}\right), \mathsf{/.f64}\left(-1, x\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y}{x}\right)\right), \mathsf{/.f64}\left(-1, x\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{\frac{1}{2} + \frac{-1}{6} \cdot y}{x}\right)\right), \mathsf{/.f64}\left(-1, x\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot y\right), x\right)\right), \mathsf{/.f64}\left(-1, x\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y\right), x\right)\right), \mathsf{/.f64}\left(-1, x\right)\right)\right)\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{6} \cdot y\right), x\right)\right), \mathsf{/.f64}\left(-1, x\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot y\right)\right), x\right)\right), \mathsf{/.f64}\left(-1, x\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(y \cdot \frac{1}{6}\right)\right), x\right)\right), \mathsf{/.f64}\left(-1, x\right)\right)\right)\right) \]

      8. *-lowering-*.f64 73.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right), x\right)\right), \mathsf{/.f64}\left(-1, x\right)\right)\right)\right) \]

   10. Simplified 73.8%

       \[\leadsto \frac{1}{x} + y \cdot \left(y \cdot \color{blue}{\frac{0.5 - y \cdot 0.16666666666666666}{x}} + \frac{-1}{x}\right) \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{6} \cdot \frac{{y}^{2}}{x}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \left(\frac{\frac{-1}{6} \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \left(\frac{\frac{-1}{6} \cdot \left(y \cdot y\right)}{x}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \left(\frac{\left(\frac{-1}{6} \cdot y\right) \cdot y}{x}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot y\right) \cdot y\right), \color{blue}{x}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(y \cdot \left(\frac{-1}{6} \cdot y\right)\right), x\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{6} \cdot y\right)\right), x\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \frac{-1}{6}\right)\right), x\right)\right)\right) \]

       8. *-lowering-*.f64 72.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right), x\right)\right)\right) \]

   13. Simplified 72.4%

       \[\leadsto \frac{1}{x} + y \cdot \color{blue}{\frac{y \cdot \left(y \cdot -0.16666666666666666\right)}{x}} \]

   if -640 < x < 5.5000000000000003e-7

   1. Initial program 81.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 81.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 83.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{1 + y}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -640:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) (+ 1.0 y))))
   (if (<= x -6.5e+203)
     t_0
     (if (<= x -640.0)
       (/ (+ 1.0 (* y (+ -1.0 (* y 0.5)))) x)
       (if (<= x 5.5e-7) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 / x) / (1.0 + y);
	double tmp;
	if (x <= -6.5e+203) {
		tmp = t_0;
	} else if (x <= -640.0) {
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / (1.0d0 + y)
    if (x <= (-6.5d+203)) then
        tmp = t_0
    else if (x <= (-640.0d0)) then
        tmp = (1.0d0 + (y * ((-1.0d0) + (y * 0.5d0)))) / x
    else if (x <= 5.5d-7) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / x) / (1.0 + y);
	double tmp;
	if (x <= -6.5e+203) {
		tmp = t_0;
	} else if (x <= -640.0) {
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / x) / (1.0 + y)
	tmp = 0
	if x <= -6.5e+203:
		tmp = t_0
	elif x <= -640.0:
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x
	elif x <= 5.5e-7:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / x) / Float64(1.0 + y))
	tmp = 0.0
	if (x <= -6.5e+203)
		tmp = t_0;
	elseif (x <= -640.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(-1.0 + Float64(y * 0.5)))) / x);
	elseif (x <= 5.5e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / x) / (1.0 + y);
	tmp = 0.0;
	if (x <= -6.5e+203)
		tmp = t_0;
	elseif (x <= -640.0)
		tmp = (1.0 + (y * (-1.0 + (y * 0.5)))) / x;
	elseif (x <= 5.5e-7)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e+203], t$95$0, If[LessEqual[x, -640.0], N[(N[(1.0 + N[(y * N[(-1.0 + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.5e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5000000000000003e203 or 5.5000000000000003e-7 < x

   1. Initial program 69.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 69.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 69.8%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{{\left(\frac{x}{x + y}\right)}^{x}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{1}}{{\left(\frac{x}{x + y}\right)}^{x}}\right)\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\left(-1 \cdot \color{blue}{x}\right)}\right)\right) \]

      8. pow-unpow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left({\left(\frac{x}{x + y}\right)}^{-1}\right)}^{\color{blue}{x}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{1}{\frac{x}{x + y}}\right)}^{x}\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x + y}{x}\right)}^{x}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\left(\frac{x + y}{x}\right), \color{blue}{x}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + y\right), x\right), x\right)\right) \]

      13. +-lowering-+.f64 69.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right), x\right)\right) \]

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \color{blue}{\left(1 + y\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 71.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \color{blue}{y}\right)\right) \]

   9. Simplified 71.9%

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

   if -6.5000000000000003e203 < x < -640

   1. Initial program 78.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 78.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)\right), x\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot y - 1\right)\right)\right), x\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot y + -1\right)\right)\right), x\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot y\right), -1\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \frac{1}{2}\right), -1\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \frac{1}{2}\right), -1\right)\right)\right), x\right) \]

   10. Simplified 68.1%

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

   if -640 < x < 5.5000000000000003e-7

   1. Initial program 81.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 81.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   7. Simplified 98.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{1}{x}}{1 + y}\\ \mathbf{elif}\;x \leq -640:\\ \;\;\;\;\frac{1 + y \cdot \left(-1 + y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.0% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -640:\\ \;\;\;\;\frac{1 + y \cdot \left(y \cdot \left(0.5 - y \cdot 0.16666666666666666\right) + -1\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -640.0)
   (/ (+ 1.0 (* y (+ (* y (- 0.5 (* y 0.16666666666666666))) -1.0))) x)
   (if (<= x 5.5e-7) (/ 1.0 x) (/ (/ 1.0 x) (+ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -640.0) {
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / x) / (1.0 + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-640.0d0)) then
        tmp = (1.0d0 + (y * ((y * (0.5d0 - (y * 0.16666666666666666d0))) + (-1.0d0)))) / x
    else if (x <= 5.5d-7) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / x) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -640.0) {
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / x) / (1.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -640.0:
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x
	elif x <= 5.5e-7:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / x) / (1.0 + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -640.0)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(Float64(y * Float64(0.5 - Float64(y * 0.16666666666666666))) + -1.0))) / x);
	elseif (x <= 5.5e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / x) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -640.0)
		tmp = (1.0 + (y * ((y * (0.5 - (y * 0.16666666666666666))) + -1.0))) / x;
	elseif (x <= 5.5e-7)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / x) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -640.0], N[(N[(1.0 + N[(y * N[(N[(y * N[(0.5 - N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.5e-7], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -640

   1. Initial program 66.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 66.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 66.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot y}\right), \color{blue}{x}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot y\right)\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right), x\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(0 - y\right)\right), x\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right), x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{0 - y}}{x}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) - 1\right)\right)}, x\right) \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y\right) - 1\right)\right), x\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + y \cdot \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) - 1\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) - 1\right)\right)\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) - 1\right)\right)\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} - \frac{1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) + -1\right)\right)\right), x\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} + \frac{-1}{6} \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{2} - \frac{1}{6} \cdot y\right)\right), -1\right)\right)\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot y\right)\right)\right), -1\right)\right)\right), x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\frac{1}{2}, \left(y \cdot \frac{1}{6}\right)\right)\right), -1\right)\right)\right), x\right) \]

      15. *-lowering-*.f64 68.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right), -1\right)\right)\right), x\right) \]

   10. Simplified 68.8%

       \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(0.5 - y \cdot 0.16666666666666666\right) + -1\right)}}{x} \]

   if -640 < x < 5.5000000000000003e-7

   1. Initial program 81.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 81.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   7. Simplified 98.8%

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

   if 5.5000000000000003e-7 < x

   1. Initial program 74.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 74.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 74.2%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{{\left(\frac{x}{x + y}\right)}^{x}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{1}}{{\left(\frac{x}{x + y}\right)}^{x}}\right)\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\left(-1 \cdot \color{blue}{x}\right)}\right)\right) \]

      8. pow-unpow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left({\left(\frac{x}{x + y}\right)}^{-1}\right)}^{\color{blue}{x}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{1}{\frac{x}{x + y}}\right)}^{x}\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x + y}{x}\right)}^{x}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\left(\frac{x + y}{x}\right), \color{blue}{x}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + y\right), x\right), x\right)\right) \]

      13. +-lowering-+.f64 74.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right), x\right)\right) \]

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \color{blue}{\left(1 + y\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 71.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \color{blue}{y}\right)\right) \]

   9. Simplified 71.9%

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

Alternative 8: 81.1% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{1 + y}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) (+ 1.0 y))))
   (if (<= x -2.5e+115) t_0 (if (<= x 5.5e-7) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / x) / (1.0 + y);
	double tmp;
	if (x <= -2.5e+115) {
		tmp = t_0;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / x) / (1.0d0 + y)
    if (x <= (-2.5d+115)) then
        tmp = t_0
    else if (x <= 5.5d-7) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / x) / (1.0 + y);
	double tmp;
	if (x <= -2.5e+115) {
		tmp = t_0;
	} else if (x <= 5.5e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / x) / (1.0 + y)
	tmp = 0
	if x <= -2.5e+115:
		tmp = t_0
	elif x <= 5.5e-7:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / x) / Float64(1.0 + y))
	tmp = 0.0
	if (x <= -2.5e+115)
		tmp = t_0;
	elseif (x <= 5.5e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / x) / (1.0 + y);
	tmp = 0.0;
	if (x <= -2.5e+115)
		tmp = t_0;
	elseif (x <= 5.5e-7)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e+115], t$95$0, If[LessEqual[x, 5.5e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000004e115 or 5.5000000000000003e-7 < x

   1. Initial program 70.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 70.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 70.2%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{1}{{\left(\frac{x}{x + y}\right)}^{x}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{1}}{{\left(\frac{x}{x + y}\right)}^{x}}\right)\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x}{x + y}\right)}^{\left(-1 \cdot \color{blue}{x}\right)}\right)\right) \]

      8. pow-unpow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left({\left(\frac{x}{x + y}\right)}^{-1}\right)}^{\color{blue}{x}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{1}{\frac{x}{x + y}}\right)}^{x}\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \left({\left(\frac{x + y}{x}\right)}^{x}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\left(\frac{x + y}{x}\right), \color{blue}{x}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(x + y\right), x\right), x\right)\right) \]

      13. +-lowering-+.f64 70.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right), x\right)\right) \]

   6. Applied egg-rr 70.1%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \color{blue}{\left(1 + y\right)}\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 70.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{+.f64}\left(1, \color{blue}{y}\right)\right) \]

   9. Simplified 70.9%

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

   if -2.50000000000000004e115 < x < 5.5000000000000003e-7

   1. Initial program 82.0%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

      3. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

      6. +-lowering-+.f64 82.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

   3. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 90.8%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

   7. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.4% accurate, 69.7× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{x \cdot \log \left(\frac{x}{x + y}\right)}\right), \color{blue}{x}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(e^{\log \left(\frac{x}{x + y}\right) \cdot x}\right), x\right) \]

   3. exp-to-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{x}{x + y}\right)}^{x}\right), x\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{x}{x + y}\right), x\right), x\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), x\right), x\right) \]

   6. +-lowering-+.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), x\right), x\right) \]

3. Simplified 75.6%

   \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 73.4%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

7. Simplified 73.4%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
8. Add Preprocessing

Developer Target 1: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{-1}{y}}}{x}\\ t_1 := \frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (/ -1.0 y)) x)) (t_1 (/ (pow (/ x (+ y x)) x) x)))
   (if (< y -3.7311844206647956e+94)
     t_0
     (if (< y 2.817959242728288e+37)
       t_1
       (if (< y 2.347387415166998e+178) (log (exp t_1)) t_0)))))

C

double code(double x, double y) {
	double t_0 = exp((-1.0 / y)) / x;
	double t_1 = pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = log(exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-1.0d0) / y)) / x
    t_1 = ((x / (y + x)) ** x) / x
    if (y < (-3.7311844206647956d+94)) then
        tmp = t_0
    else if (y < 2.817959242728288d+37) then
        tmp = t_1
    else if (y < 2.347387415166998d+178) then
        tmp = log(exp(t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-1.0 / y)) / x;
	double t_1 = Math.pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = Math.log(Math.exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-1.0 / y)) / x
	t_1 = math.pow((x / (y + x)), x) / x
	tmp = 0
	if y < -3.7311844206647956e+94:
		tmp = t_0
	elif y < 2.817959242728288e+37:
		tmp = t_1
	elif y < 2.347387415166998e+178:
		tmp = math.log(math.exp(t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-1.0 / y)) / x)
	t_1 = Float64((Float64(x / Float64(y + x)) ^ x) / x)
	tmp = 0.0
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-1.0 / y)) / x;
	t_1 = ((x / (y + x)) ^ x) / x;
	tmp = 0.0;
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision]}, If[Less[y, -3.7311844206647956e+94], t$95$0, If[Less[y, 2.817959242728288e+37], t$95$1, If[Less[y, 2.347387415166998e+178], N[Log[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -37311844206647956000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (exp (/ -1 y)) x) (if (< y 28179592427282880000000000000000000000) (/ (pow (/ x (+ y x)) x) x) (if (< y 23473874151669980000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x)))))

  (/ (exp (* x (log (/ x (+ x y))))) x))